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## Material Information- Title:
- Type I error probabilities and power of the rank and parametric ancova procedures
- Creator:
- Seaman, Samuel L., 1954-
- Publication Date:
- 1984
- Language:
- English
- Physical Description:
- v, 76 leaves : ; 28 cm.
## Subjects- Subjects / Keywords:
- Applied statistics ( jstor )
Correlations ( jstor ) Covariance ( jstor ) Educational research ( jstor ) Experimental procedures ( jstor ) False positive errors ( jstor ) Mathematical dependent variables ( jstor ) Sample size ( jstor ) Skewed distribution ( jstor ) Statistical discrepancies ( jstor ) Analysis of covariance ( lcsh ) Dissertations, Academic -- Foundations of Education -- UF Foundations of Education thesis Ph. D Probabilities ( lcsh ) Statistical hypothesis testing ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1984.
- Bibliography:
- Bibliography: leaves 71-75.
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Samuel L. Seaman.
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- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 030585935 ( ALEPH )
11940929 ( OCLC )
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REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID EZ10DIVQU_HQ55IU INGEST_TIME 2014-07-02T22:18:02Z PACKAGE AA00022723_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES PAGE 1 TYPE I ERROR PROBABILITIES AND POWER OF THE RANK AND PARAMETRIC ANCOVA PROCEDURES By SAMUEL L. SEAMAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1984 PAGE 2 To Aaron Phillip PAGE 3 ACKNOWLEDGMENTS I am indebted to Dr. Gordon G. Bechtel, Dr. Linda M. Crocker, and Dr. Stephen F. Olejnik for their many valuable suggestions which helped to shape this dissertation. I am especially grateful to Dr. James J. Algina for his unending patience and editorial assistance on the many editions of this manuscript. I thank my wife, Mary, for typing the manuscript, and for her patience and understanding during its' preparation. Finally, I thank my parents for their constant love and support. S.L.S ii PAGE 4 TABLE OF CONTENTS ACKNOWLEDGMENTS ABSTRACT CHAPTER I INTRODUCTION The Problem Purpose of the Study Significance of the Study II REVIEW OF RELATED LITERATURE Overview Parametric vs . Nonparametric ANCOVA Summary , III METHODOLOGY , Design , Condition Combinations Summary IV RESULTS AND DISCUSSION Type I Error Probabilities Power Differences Homogeneous Conditional Variances Heterogeneous Conditional Variances V CONCLUSIONS BIBLIOGRAPHY BIOGRAPHICAL SKETCH PAGE ii iv 4 7 11 13 13 19 24 27 27 33 36 38 38 44 45 55 65 71 76 in PAGE 5 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy TYPE I ERROR PROBABILITIES AND POWER OF THE RANK AND PARAMETRIC ANCOVA PROCEDURES By Samuel L. Seaman August, 1984 Chairman: James J. Algina Major Department: Foundations of Education The purpose of the present study was to examine the Type I error rates and statistical powers for the parametric and rank analysis of covariance procedures for a variety of experimental conditions that have not been considered in previous research. The probability of obtaining a significant statistic, using the parametric and rank analysis of covariance procedures, was estimated for a number of experimental conditions defined in terms of the conditional distributions for two groups in a randomized experiment. The two conditional distributions were both normal, both skewed to the same degree, but in different directions, or both skewed in the same direction but to different degrees . Two ratios of conditional variance were investigated for each of the three possibilities described above. A ratio of one: one was used to simulate the experimental condition iv PAGE 6 of homogeneous conditional variances. A ratio of two: one was used to simulate a modest degree of between group heterogeneity of variance. For each type of pair of conditional distributions, both equal and unequal conditional means were studied. In addition, sample size and size of conditional variance were also manipulated. Results were interpreted in terms of Type I error probabilities and statistical power for testing two different null hypotheses , equality of two conditional means, or equality of two conditional distributions. It was concluded that the parametric analysis of covariance procedure is to be preferred as a test of both hypotheses provided that the conditional distributions are normal, or the conditional distributions are non-normal and the correlations among height of conditional mean function, degree (or direction) of skew, and size of conditional variance are expected to be positive. For other combinations of these correlations , the choice of a statistical test becomes much more complicated, and depends upon the sample size, size of conditional variance, form of conditional distributions, and hypothesis to be tested. In most of the cases studied, however, the choice of a particular statistical test was inconsequential and a general use of the parametric procedure was indicated. v PAGE 7 CHAPTER I INTRODUCTION The analysis of covariance is often used to analyze data obtained in educational research studies conducted using the pretest-posttest control group design (Campbell and Stanley, 1963). In the analysis, the pretest is treated as the covariate and represents a source of variation that is expected to be correlated with the dependent variable. The posttest is treated as the dependent variable. When used with a randomized experiment, which is the context assumed in this dissertation study, the purpose of analysis of covariance is to increase power and efficiency relative to that obtained using an analysis of variance of the posttest data. The linear model for a one-way, fixed effects design with one dependent variable and one covariate is given by Y ij =^+Â«j + /Vx ij ) +Â£ij where Y is the i th observation for the i th treatment on the dependent variable, X is the covariate for subject i in treatment level j, m. is the constant for overall mean response conditional upon X, <*. is the constant for the PAGE 8 j treatment effect, /S w is the linear regression coefficient of Y on X, and Â£^. axe random variables assumed to be normally and independently distributed with mean zero and common variance. The assumptions for analysis of covariance are well known (see for example, Huitema, 1980); however, they are presented here for completeness: a) The regression slopes associated with the various treatment groups are equal. b) The relationship between the covariate and dependent variable is linear. c) Within each group the distribution of dependent variable scores conditional on the covariate is normal. d) For each value of the covariate measure, the variances conditional on the covariate are homogeneous both within and between treatment groups . The latter two assumptions, conditional normality and homogeneity of variance, are most pertinent to the following discussion. These four assumptions must be met for the F ratio, calculated in the analysis of covariance procedure, to be distributed as the F distribution. In practice, it is unlikely that all of the assumptions of the analysis are met exactly. If one or more of the assumptions are violated, the researcher may decide to proceed with the PAGE 9 analysis at some risk, or abandon the model and use instead an alternative procedure that does not require all of the assumptions of the parametric analysis of covariance model. Several nonparametric analysis of covariance strategies have been reported in the literature (Quade, 1967; Puri and Sen, 1969; McSweeney and Porter, 1971; Burnett and Barr, 1977; Shirley, 1981). These procedures do not require assumptions b , c or d given above . If the researcher does not have a way of determining what effect violations of the assumptions might have on his analysis , then interpretation of the results may be inaccurate (Zikri, 1983). It is not surprising then, that a number of research studies have considered the robustness properties and statistical power of the parametric analysis of covariance model when assumptions have been violated. (Atiqullah, 1964; Levy, 1980; McClaren, 1972; Peckham, 1970; Rogosa, 1980; Shields, 1978; Thomson, 1980; Wildemann, 1974) . Additional studies have compared these same properties for the parametric analysis of covariance and one or more of the nonparametric alternatives (Abunnaja, 1983; Conover and Iman, 1982; McSweeney and Porter, 1971; Olejnik and Algina, 1983, 1984; Olejnik, Algina and Abdel-Fattah, 1984). In the present study, the type I error rates and statistical powers of the parametric and rank analysis of covariance procedures were examined PAGE 10 under violations of the conditional normality and/or homogeneity of variance assumptions . The Problem Huitema (1980) in summarizing much of the research in the area reports that, when sample sizes are equal, parametric analysis of covariance may not be drastically affected by violations of the assumptions of conditional normality, and between or within group homogeneity of conditional variances. If, however, the assumptions of normality and/or between group homogeneity of conditional variances, are seriously violated, especially when sample sizes are unequal, the parametric analysis of covariance procedure may yield biased F ratios . Olejnik and Algina (1984) reviewed the alternative nonparametric analysis of covariance procedures. They provided a detailed discussion of each procedure, and comparative information relative to type I error rates and statistical power. Olejnik and Algina found that, when the parametric analysis of covariance is robust to simulated violations of the assumptions of conditional normality and between and within group homogeneity of variance, the nonparametric alternatives suggested by Quade, Puri and Sen, and McSweeney and Porter also have acceptable type I errors. (Their review also suggests that the Burnett and Barr, and Shirley procedures frequently yield unacceptable type I errors and should not be considered viable alternatives to parametric analysis of PAGE 11 covariance) . Moreover, even when extreme non-normality results in discrepancies between actual and nominal type I error rates for the parametric analysis of covariance, the Quade, Porter and McSweeney and Puri and Sen procedures tend to yield appropriate type I error rates. Olejnik and Algina did not review any investigations of violations of between group homogeneity either alone or in combination with conditional nonnormality, and the literature review conducted for this dissertation did not uncover any such studies. A number of researchers have concluded that when distributional assumptions are violated, the nonparametric procedures should be used in place of their parametric counterparts even when the parametric procedures are robust to violations of those assumptions. Blair and Higgins (1980), while discussing the Wilcoxin test as an alternative to the parametric t-test, make this recommendation. They cited the tendency for the nonparametric procedure to be more powerful than the parametric procedure for a wide variety of non-normal distributions. And, McSweeney and Porter (1971) , in discussing their nonparametric analysis of covariance offer this advice: The fact that the relative advantage of the parametric tests was slight even when the assumptions necessary for their valid use were completely satisfied suggests that little loss and possible considerable gain, in power will result from the more general use of these nonparametric analouges. (p. 39) PAGE 12 McSweeney and Porter's suggestion that their nonparametric approach might prove more powerful for a variety of non-normal conditional distributions has since been substantiated by Abunnaja (1983), Conover and Iman (1982), Olejnik and Algina (1983, 1984), and Olejnik, Algina and Abdel-Fattah (1984) . A common characteristic of all the investigations comparing the nonparametric alternatives to the parametric analysis of covariance is that each considered situations in which there were no between group differences in the conditional distributions. The only between group differences examined were differences in the conditional means . Thus their results do not provide guidance for the researcher who must choose between parametric and nonparametric analysis of covariance procedures in situations where between group differences in skew and/or variance exists alone or in conjunction with between group differences in conditional means . The summary of the literature presented above suggests a void concerning the comparison of parametric and nonparametric analysis of covariance. In the first place, nothing is known about how nonparametric alternatives to analysis of covariance function when there is between group heterogeneity of variance. Secondly, nothing is known about how the parametric analysis of covariance or the nonparametric alternatives function when the conditional distributions differ, between groups, in skew. PAGE 13 7 The general purpose of this study was to examine properties of the parametric analysis of covariance and a nonparametric alternative for the experimental conditions characterized above. It was felt that results of the present study would then provide the necessary information to fill this void. The rank analysis of covariance was chosen as the nonparametric analog in this study since it is computationaly more convenient than and provides power estimates similar to Quade's and Puri and Sen's procedures. These two procedures are the only other nonparametric analysis of covariance procedures that tend to yield appropriate Type I errors (Olejnik and Algina, 1984). In the rank transform procedure the dependent variable and covariate scores are transformed by ranking the original scores without regard to treatment groups. The transformed data are then analyzed using the usual parametric analysis. The test statistic is referred to the F distribution with the same degrees of freedom as the parametric case. Purpose of the Study Most research designs for comparing the effectiveness of educational treatments yield data which can be analyzed by both parametric and nonparametric tests . In making a choice between the parametric analysis of covariance procedure and the nonparametric rank analysis of covariance procedure it would be useful to have estimates of the actual probabilities of type I error and the actual powers of the PAGE 14 8 two tests for a variety of experimental conditions where the assumptions of conditional normality and/or homogeneity of variance have been violated. Given this information and a knowledge of the distributions encountered in their studies, researchers can choose the most powerful test from those that are known to yield actual type I error rates near the nominal alpha level. Such tests are said to preserve their level. Recently there have been a number of investigations (Abunnaja, 1983; Conover and Iman, 1982; Olejnik and Algina, 1983, 1984; Olejnik, Algina and Abdel-Fattah, 1984) of parametric and nonparametric analysis of covariance which provided information of the kind described above. The nonparametric analysis of covariance procedure employed in these studies was the rank analysis of covariance procedure suggested by McSweeney and Porter (1971). Abunnaja (1983) and Olejnik and Algina (1984) also examined the nonparametric alternatives suggested by Burnett and Barr (1977) , Puri and Sen (1969) and Quade (1967) . A comprehensive review of these studies will be provided in the next chapter, but of interest now are the general findings. Taken together, the results of these studies suggest that for most experimental conditions, both the parametric and rank analysis of covariance procedures tend to preserve their levels for a wide variety of non-normal conditional distributions. In terms of power, however, the parametric procedure enjoys PAGE 15 significant power advantages in certain situations while the rank procedure provides a more powerful test in other situations. While the Abunnaja, Conover and Iman, and Olejnik, Algina and Abdel-Fattah studies provide useful information for the researcher who must choose between the parametric and rank of covariance procedures, they do not address two important situations which have become the logical motivation for this study. A first consideration was to examine the type I error rates and powers of parametric analysis of covariance and rank analysis of covariance when they are used to test the hypothesis of equality of conditional means, and the conditional distributions for the groups differ in scale and/or shape. Here, preservation of the level of the parametric analysis of covariance is in question because the conditional distributions are non-normal, and robustness to this kind of non-normality has not been examined by previous research in the field. Preservation of the level of the rank analysis of covariance procedure is questionable since the researcher is interested only in detecting differences in conditional means, and the rank analysis of covariance procedure may be sensitive to between group differences in shape, scale, or shape and scale (Savage, 1962) A second purpose was to examine statistical powers of the parametric and rank procedures when used to test the hypothesis of equality of conditional distributions, and PAGE 16 10 those conditional distributions may differ in location and scale, location and shape, or location, scale, and shape. With regard to preservation of level in this case, the work of Abunnaja, Conover and Iman, and Olejnik and Algina has shown that if the joint covariate-dependent variable distributions are the same for both groups , then over a wide range of non-normal conditional distributions both the rank and parametric analysis of covariance procedures tend to yield actual alpha levels near the nominal alpha levels . (An exception did occur in the Conover and Iman (1982) study where the parametric analysis of covariance procedure proved to be somewhat conservative with extremely non-normal conditional distributions.) In this case then, the general concern of this study was to examine the comparative powers of the rank and parametric analysis of covariance procedures when they are used to test the hypothesis of equality of conditional distributions , and it is expected that those distributions differ in location and shape, location and scale, or location, shape and scale. The two cases described above were investigated using Monte Carlo simulation procedures . The scenario was a randomized experiment comparing two experimental groups . For all experimental conditions simulated, the scale and shape of the conditional distributions were independent of the covariate scores and a covariate by treatment interaction did not exist. As a result, conclusions about the utility PAGE 17 11 of the parametric and rank analysis of covariance procedures for testing hypotheses about conditional means and conditional distributions also apply to testing the corresponding hypotheses about marginal means and marginal dependent variable distributions. Significance of the Study Campbell and Stanley (1963) express great concern for validity in experimentation: Internal validity is the basic minimum without which any experiment is uninterpre table: Did in fact the experimental treatments make a difference in this specific experimental instance? (pg. 5) The analysis of covariance is often used in educational research to answer this question. Kirk (1982) identifies the advantages of using the analysis of covariance procedure: a) There is a reduction in experimental error and, hence, an increase in statistical power. b) Bias caused by differences among experimental units is reduced when those differences are not attributable to the manipulation of the independent variable. The advantages given above may not exist when assumptions underlying the model have not been met. If one or more violations are suspected, a nonparametric alternative may be the more appropriate choice. The validity of the conclusion obtained using either procedure, parametric or nonparametric, will certainly be affected by the researchers' PAGE 18 12 knowledge of the effect that the violations had on the final results of their analyses. Studies examining the type I error rates and statistical powers of the parametric and nonparametric analysis of covariance procedures have provided researchers with empirical evidence useful in making this choice for a number of experimental conditions. The present study examines similar properties of the two procedures for additional experimental conditions where the assumptions of conditional normality and/or between group homgeneity of variance have been violated in a distinctive way. Results of the study will provide additional guidelines that should prove useful in making an astute choice between the parametric and nonparametric statistical procedures, thereby enhancing the probability of accurate and valid conclusions in educational experimentation. PAGE 19 CHAPTER II REVIEW OF RELATED LITERATURE Overview In a published review of the analysis of variance and analysis of covariance procedures, Glass, Peckham, and Sanders (1972) summarized the research that had examined the effects of violations of the assumptions of fixed effects analysis of variance and covariance. While much of their article dealt with the analysis of variance procedure, a number of studies that had investigated the analysis of covariance procedure were also reviewed. Some of the earlier studies cited, suggested that the analysis of covariance was robust to violations of the assumptions of normality and homogeneity of variance (Cochran, 1957; Winer, 1962). The analysis of covariance was shown, analytically, to be insensitive to violations of the normality assumption when the covariate is approximately normally distributed (Box and Anderson, 1962; Atiquallah, 1964). Potthoff (1965) suggested that the sensitivity of the analysis of covariance to heterogeneity of conditional variances depended on the ratio, n lÂ°xl / n 2^x2 PAGE 20 14 This would suggest that the sensitivity of the analysis of covariance to heterogeneity of conditional variances depends upon the covariate , X, and sample size as well. Peckham (1968) concluded that inequality of regression slopes, unless extreme, had little effect on the analysis of covariance procedure. Atiquallah (1964), however, had suggested that the analysis of covariance was insensitive to unequal regression slopes only if the means of the concomitant variables are equal or the within group sums of squares are equal. Finally, based on several investigations examining unreliable covariate measures (Cochran, 1968; Lord, 1960; Porter, 1967), Glass et al. conclude that, as errors of measurement in X increase, the analysis becomes more like the corresponding analysis of variance. Thus, (1) the increase in precision afforded by ANCOVA is attenuated and (2) there is less reason to assume that "the groups have been statistically equated" on the concomitant variable, (pg. 281) In the wake of Glass, Peckham, and Sanders' (1972) article, numerous investigations of the effects of violation of the assumptions of covariance analysis have been conducted. Hamilton (1976, 1977) was particularly interested in the assumption of homogeneity of regression slopes and a previous study that had been performed by Peckham (1968). In particular, Hamilton (1977) was interested in the effect that unequal regression slopes might have not only on type I errors, but also PAGE 21 15 on the power of covariance analysis. Hamilton found that the analysis of covariance was apparently robust to violations of homogeneity of regression when group sizes were equal. When group sizes were unequal, and the homogeneity of regression assumption was violated, large discrepancies between actual alpha levels and corresponding nominal alpha levels were observed. In terms of power, when sample sizes were equal, violation of homogeneity of regression had little effect on the statistical power. In another empirical investigation, Levy (1980) examined the effects of unequal regression slopes and/or conditional non-normality. Levy's results suggest that the analysis of covariance procedure is robust to violations of the homogeneity of regression slopes and/or conditional normality when sample sizes are equal. The analysis of covariance procedure also appeared to be robust to violations of conditional normality, whether or not sample sizes were equal, so long as the regression slopes are equal. When assumptions of conditional normality and homogeneity of variance were violated simultaneously, and sample sizes were unequal, the actual alpha levels for the analysis of covariance procedure began to deviate significantly from the nominal alpha levels . A number of recent analytical studies investigating the effect of heterogeneity of regression slopes PAGE 22 16 (Hollingsworth, 1980; Rogosa, 1980) provide contradictory conclusions but in general suggest that unequal slopes produce biased test statistics. In addition, Hollingsworth (1980) concluded that statistical power is also affected by heterogeneity of regression slopes. Hsu (1983) is quick to point out however, that practical significance of the bias produced by assumption violations cannot be determined analytically, and suggests that despite the obvious problems associated with Monte Carlo studies, they should in fact be carried out to supplement analytical studies. Finally, Abunnaja (1983) examined the effects of violation of conditional normality and/or homogeneity of regression, again with equal and unequal sample sizes. Abunnaja extended the Levy (1980) study by considering, as well, several nonparametric alternatives to the parametric analysis of covariance. The results of Abunnaja' s study are consistent with those of Levy, suggesting that the parametric analysis of covariance is robust to the violations singly and in combination, so long as sample sizes are equal and the differences in regression slopes are not extreme. The nonparametric procedures were also found to preserve their levels for the violations examined in the study. Earlier in this discussion it was suggested that the assumptions most pertinent to the present study were the assumptions of conditional normality and PAGE 23 17 homogeneity of conditional variances. A number of studies have examined the effects of violating one or both of these assumptions, and are discussed in detail below. Wildemann (1974) conducted a Monte Carlo study in which she investigated the effects on the covariance analysis of a conditional non-normality resulting from the use of measuring instruments that yield only discrete observations on a limited score scale. Her results suggest that covariance analysis preserves its level for varying degrees of skewness or discreteness in the covariate and/or dependent variable. Another Monte Carlo investigation conducted by Shields (1978) examined the effect of violating the assumptions of homogeneity of conditional variances on the analysis of covariance and the Johnson-Neyman technique. The homogeneity of variance assumption may be violated in two ways (Elashoff, 1969). The first form of violation occurs when the variance of the dependent variable scores is constant across treatment groups for all values of the covariate, but there is a distinct relationship between the covariate and the variance of the dependent variable (within group heteroscedasticity) . The second form of this violation occurs when the group variances on the dependent variable are different, but within each group, the variances of the dependent variable are constant for all values of the covariate (between group heteroscedasticity) . PAGE 24 18 In her study Shields simulated both forms of the violation and investigated the effects such violations had on type I error rates. Her results suggest that the analysis of covariance is robust to the violations of assumptions of within group homoscedasticity (form 1) and/or between group homogeneity (form 2) of variance, provided sample sizes are equal. When sample sizes differed, violation of the assumption of within group homoscedasticity of conditional variances had little effect on type I errors. For different sample sizes a violation of the between group homoscedasticity of variance assumption, however, resulted in unacceptable actual alpha levels when compared to the nominal alpha levels. When sample size and variance were positively correlated, the analysis provided a conservative hypothesis test. For a negative correlation between sample size and variance, the hypothesis test was liberal. Shields did not consider the effects that heteroscedasticity and/or heterogeneity had on the power of the statistical procedures. In a similar study, Zikri (1983) examined the effect of violation of the assumptions of between group homogeneity of conditional variances and/or conditional normality. The results of Zikri' s study parallel those of Shields. When sample sizes were equal, the analysis of covariance procedure was robust to violation of the conditional normality and/or homogeneity of variance assumptions. When sample sizes were unequal, the same test biases were PAGE 25 19 found as in Shields' study. Zikri also considered the effects violations of the above assumptions had on statistical power, when actual alpha levels were found to be appropriate. When sample sizes were equal and covariate means were equal, violation of the conditional normality and/or homogeneity assumptions had no real affect on statistical power of the analysis of covariance procedure. If sample sizes differed, or if the covariate means were unequal, the power of the statistical procedure was found to be greatly reduced for a large number of the simulated conditions. Thomson (1980) conducted an empirical investigation into the effect of heterogeneity of within-cell population regression coefficients and heterogeneity of within-cell population error variances , on type I error rates of the analysis of covariance procedure. Again, when sample sizes were equal, the violations had little effect on type I errors. If sample sizes are unequal, the violation of homogeneous variances results in aberrant actual alpha levels when compared to the nominal alpha levels. Parametric vs. Nonparametric ANCOVA As indicated earlier, the general purpose of the present study was to provide guidelines which might assist the researcher who must choose between the parametric analysis of covariance or an alternative nonparametric alternative. Studies by Abunnaja (1983) and Olejnik and PAGE 26 20 Algina (1984) were cited earlier. The general findings of these studies suggested that of the common nonparametric alternatives, those offered by McSweeney and Porter, Puri and Sen, and Quade were found to preserve their levels for a wide range of non-normal conditional distributions . It was further noted that the procedure suggested by McSweeney and Porter was computationally the easiest procedure to implement for a given data set, and had power estimates quite similar to the other acceptable procedures . The focus of the following literature review is therefore on studies which have compared the parametric analysis of covariance procedure and at least one of its nonparametric analogs. A study by Abunnaja (1983) cited earlier in a discussion of the effect of violation of the homogeneity of regression assumption, also considered the effects of conditional non-normality and the combined effects of conditional non-normality and heterogeneity of regression, for the parametric analysis of covariance and a number of nonparametric alternatives . The results of the study suggested that both the parametric and three of the nonparametric procedures had acceptable type I errors for conditions where conditional normality and/or homogeneity of regression assumptions had been violated. The three nonparametric procedures were those suggested by McSweeney and Porter, Puri and Sen, and Quade. PAGE 27 21 Further, the results suggested that the three nonparametric procedures cited above had comparable empirical power, but their power values were consistentlysmaller than powers of the parametric procedures examined. Conover and Iman (1982) compared the parametric analysis of covariance and rank analysis of covariance procedures by simulating data for the situation in which four treatments were to be compared. The conditional distribution for each group was normal, lognormal, exponential, uniform or Cauchy. In the null case the joint covariate-dependent variable distributions were the same for all four groups. The level of the rank procedure was preserved for all five distributions. The level of the parametric procedure was preserved with the normal, exponential and uniform distributions, but was conservative with the lognormal and Cauchy distributions. In the non-null situation the covariate distributions were the same for the four groups but the conditional distributions differed in location for five of the six pairs of groups. The parametric ANCOVA was more powerful with the normal and uniform distributions but less powerful with the exponential, lognormal, and Cauchy distributions. Olejnik and Algina (1983) conducted a simulation which focused on the two-group situation. Like Conover and Iman they studied the effects of nonnormal conditional distributions. However, the degree of PAGE 28 22 conditional non-normality was much less extreme than in the Conover-Iman study. In addition, Olejnik and Algina investigated the effect of within group heteroscedasticity and the combined effects of conditional non-normality and within group heteroscedasticity. In the null case the joint covariate-dependent variable distributions were the same for both groups, while in the non-null case only the conditional means differed across groups. Olejnik and Algina found that the level of both the rank and parametric ANCOVA was preserved with non-normal, heteroscedastic, and non-normal and heteroscedastic conditional distributions. They also found that when the conditional distributions differed in location the rank procedure can have a small advantage if the conditional distributions are leptokurtic and the covariate-dependent variable correlation is small (p= 3 in their study). In addition the rank ANCOVA can have a substantial power advantage if the conditional distributions are leptokurtic, skewed and heteroscedastic, skewed and leptokurtic, or skewed, leptokurtic and heteroscedastic. The parametric ANCOVA can have a small power advantage if the conditional distribution is skewed and the covariate-dependent variable correlation is large (/* -7 in their study). Olejnik, Algina and Abdel-Fattah (1984) examined the effects that violations of conditional normality and/or within group homogeneity of variance had on the statistical PAGE 29 23 power of the parametric analysis of covariance procedure and on the differences in statistical power between the parametric and rank analysis of covariance procedures. Results of their study suggest that the statistical power of the parametric analysis of covariance procedure is little affected by modest violations of the conditional normality and/or within group homogeneity of variance assumptions. The statistical power of the rank analysis of covariance procedure, however, was found to increase with the introduction of conditional nonnormality and/or within group heterogeneity of variance. The greatest increases were observed for conditions where the relationship between the covariate and posttest measures was relatively weak. Olejnik, Algina and Abdel-Fattah also reported that the strength of the covariate-dependent variable relationship had a significant effect on the statistical powers of the two procedures (parametric and rank ANCOVA) . They found that when the covariate-dependent variable relationship was moderate or stronger (/3 s . 7), there was little reason to use the rank analysis of covariance procedure, as power advantages, when they did exist, were minimal. When the covariate-dependent variable relationship was weak ( / = .3) ) however, the rank analysis of covariance procedure did enjoy power advantages over the parametric analysis of covariance. Olejnik, Algina and Abdel-Fattah also developed power curves, and examined the power differences over the entire range of the power curves. They found that the power differences between the parametric and rank analysis of covariance PAGE 30 24 procedures were not consistent over the entire range of each power curve. The largest differences in power occurred for the most part, at the middle of the power curve (parametric powers ranging from .4 to .6). A general conclusion derived in their study was that power differences (parametric vs. rank analysis of covariance) were a function of the strength of the covariate-dependent variable relationship, the degree to which assumptions were violated, and the statistical power of the parametric analysis of covariance. Summary The research findings reviewed above suggest that when sample sizes are equal, the analysis of covariance procedure tends to be robust to moderate violations of a number of the usual assumptions both singly and in combination. In those studies where a nonparametric analysis of covariance procedure was examined, the nonparametric procedure too, yielded actual alpha levels near the nominal alpha for a variety of violations of assumptions. When unequal sample sizes were encountered, however, the parametric procedure was found to yield actual alpha levels that were significantly different from the nominal alpha levels . In terms of statistical power, research findings reviewed provide a number of interesting conclusions. When violations of the conditional normality assumption are extreme, the rank analysis of covariance usually enjoys significant power advantages over the parametric analysis of covariance procedure. For less extreme violations of the conditional normality assumption, smaller power advantages associated PAGE 31 25 with the rank analysis of covariance were found when the covariate-dependent variable correlation was weak. If the covariate-dependent variable relationship is large the parametric procedures tend to enjoy small power advantages over the rank procedure. Finally, considerable power advantages favoring the rank ANCOVA were observed for the combined violations of the conditional normality and homogeneity of within group variance assumptions. These differences were found to increase with any associated increase in sample size. However, the power advantages were shown to be greatest for the central portion of the power curves . A common characteristic of all investigations comparing the nonparametric and parametric procedures was that each study considered situations where no between group differences in conditional distributions existed. The only between group differences, were differences in conditional means. The results of these studies, therefore, do not provide guidance for the researcher who must choose between the parametric and nonparametric analysis of covariance procedures in situations where between group difference in skew and/or variance exist alone or in conjunction with between group differences in conditional means. The general purpose of this study then was to examine properties of the parametric analysis of covariance and a nonparametric analog for the unique experimental conditions characterized above. Results of the present study should PAGE 32 26 then provide the necessary information to fill the void that now exists, enabling the researcher to make an astute choice of a statistical procedure in situations where between group differences in skew and/ or variance are expected. PAGE 33 CHAPTER III METHODOLOGY In the present study, data for a one-way fixed effects analysis with two treatment groups, one dependent variable, and one covariate, were simulated. Four factors were of interest: a) sample size, b) size of the conditional variance, c) form of the conditional distribution, and d) heterogeneity of conditional variances. The levels of these factors were combined so that a number of experimental conditions could be simulated. The specific levels of the factors examined, the method of data generation, and combinations of all levels investigated are described in detail below. Design Sample Size. The study involved comparisons of two groups having equal sizes of either 20 or 40 observations in each. A cell frequency of 20 was seen as being representative of a small experiment, with smaller cell sizes occurring infrequently in educational research. A cell frequency of 40 was chosen as being representative of moderate sized experiments, and more likely to occur in educational studies than would two group experiments with 27 PAGE 34 28 larger cell frequencies. Numerous research studies have documented the detrimental effects of conducting experiments where sample sizes were unequal, the general consensus being that practitioners should avoid unequal sample sizes. The effect of inequality of sample sizes was therefore not considered in the present study. Size of the Conditional Variance . Two levels of conditional variance were investigated, 1 .3 2 and 1 .7 . In the homoscedastic case these conditional variances resulted in covariate-dependent variable correlations of .7 and .3 respectively. A correlation of .3 represents a relatively weak covariate-dependent variable relationship, while the correlation of .7 represents a relatively strong relationship. Correlations encountered in behavioral science research often occur in the interval from .2 to .7, and hence the correlations above reflect realistic values for this parameter. In the heteroscedastic case the average conditional variance was either 1 .3 2 or 1 .7 2 . This assured that any effect of heteroscedasticity was not confounded with the size of the error term. Form of the Conditional Distributions . The conditional distributions for the two treatment groups were either both normal or both skewed. For the conditions involving two skewed distributions, there were two distinct cases PAGE 35 29 to be considered. In one case, hereafter referred to as Case A, the degree of skewness was the same in both groups, but positive for one and negative for the other. For Case A, two levels of skewness were investigated; + .25, and + .75. Relative frequency distributions based on a random sample of 10,000 observations from distributions with skewness .25 and .75 respectively are presented in Table 1. The levels of skewness were chosen to reflect very small to moderate departures from normality. It was felt that if in practice conditional distributions differ in sign of skew, the degree of skew would be fairly small for behavioral data. A second case, hereafter referred to as Case B, considered two groups having conditional distributions that were skewed in the same direction, but had degrees of skewness that were different. Here, two different combinations of skew were considered, .25/. 50, and .25/. 75. A relative frequency distribution based on a random sample of 10,000 observations from a distribution with skewness .50 is also presented in Table 1. The first combination represents groups having mildly skewed conditional distributions that differ only slightly in degree of skewness. The second combination represents groups having slightly skewed and moderately skewed distributions that now differ significantly in degree of skew. Again, it was felt that these combinations PAGE 36 30 Table 1 Relative Frequency Distribution for Observations of Three Skewed Random Variables Standard Deviations 3 Degree of Skew from the Mean .25 50 75 -co, -3 .0 -3 Â• 0, -2 .0 -2 Â• 0, -1. .0 -1 0, 0, .0 .0, 1. ,0 1. o, 2. 2, o, 3. 3. o, oo 1.52 14.34 17.05 18.20 36.16 36.91 37.82 31.75 29.57 26.58 13.40 13.25 13.34 2.57 2.87 3.67 .26 .35 .39 Each frequency distribution is based on 10,000 observat ions PAGE 37 31 would reflect conditions that are most likely to occur in practice. Ratio of Conditional Variances . Two ratios of conditional variances were considered. The first, a ratio of 1, simulates the experimental condition of homogeneous conditional variances. The second, a ratio of 2, simulates the experimental condition where a modest degree of heteroscedasticity exists. Shields (1978) examined the effect of more severe degrees of heteroscedasticity on the robustness of the parametric analysis of covariance. Again, in the present study experimental conditions likely to occur in actual practice were of primary concern, and hence it was more important to examine the effects of a more modest degree of heteroscedasticity. Generation of the Data . The equation used to generate the dependent variable data is given by Y=/5j X + E (Sj)/l -/>j i + C y (2) For experimental conditions where the conditional mean functions were the same for both groups an initial data set was generated by allowing Cj to equal the constant in both groups. This initial data set was then used to generate data for the non-null case by adding an appropriate constant, C 2 , to each of the scores in group 2. Hence, C 2 was set equal to .2, .4, .6, or .8 when the sample size was 20 in each group (.15, .30, .45, or .60 when the sample size was 40 in each group) , so that the statistical powers could be compared for various effect sizes. In all PAGE 38 32 experimental conditions simulated, X was distributed normally with mean 50 and variance 1. To simulate those experimental conditions where conditional variances were equal, S. was set equal to 1 for both groups. When conditional variances differed for the two groups, one of the Sj was set equal to 1 while the other S^ was set equal tOy/2. Finally, for the homoscedastic case P j was set equal to .3 or .7 simulating the two levels of conditional variance discussed previously. For the heteroscedastic case fj was set equal to .6271629 or .8124038 simulating the two levels of average conditional variance. The normally distributed covariate data were simulated by generating normal random variables using the Statistical Analysis System (1982) normal function, and adding the constant 50 to each. Again, normal conditional distributions were simulated by generating normal random variables and assigning these values to E in (2) . The non-normal conditional distributions were simulated by transforming a standard normal variable (z) to a new variable with mean zero, variance one, and a known degree of skewness. The degree of skewness was manipulated by choosing the appropriate coefficients in the formula E = c + bz + cz 2 + dz 3 given by Fleishman (1978) . For each of the experimental conditions studied, observations were simulated from two populations. The simulations were replicated 1000 times for each experimental condition, and the resulting data were analyzed using the analysis of covariance and the rank analysis of covariance procedures . PAGE 39 33 Conditional Combinations Conditional Normality . When the conditional distributions were homoscedastic, there were four possible combinations of sample size and the size of conditional variance, and data were simulated for each of these possible combinations. When the conditional distributions were heteroscedastic, it was possible to have either a positive or negative correlation between C, and S, . Each of these correlations could have been combined with the four combinations of sample size and size of the conditional variance, for a total of eight additional conditions. However, data simulating the positive correlation can be converted to data simulating the negative correlation by the transformation Y* = -Y. Obviously, the analysis of Y* and Y will yield the same results and it was unnecessary to simulate both patterns of correlations. Case_A. For the heteroscedastic conditional distributions, there were eight possible patterns of positive and negative correlations among C j , Sj , and sign of skew. However, since the data were simulated for only two groups, once the correlations between two pairs of these variables were established, the third correlation was automatically determined. In this present study, then, there were really only four distinct patterns of correlations among the three variables. Moreover, if the sign of skew and C. had a positive correlation, then the two possible patterns of correlations of these variables with Sj involved variables PAGE 40 34 related by the function Y* = -Y. Similarly if sign of skew and Cj had a negative correlation then the two possible patterns of correlations of these variables with Sj again involved variables related by the function Y* = -Y. As a result only the correlation between sign of skew and Cj was critical, and only one of the two possible conditions involving a positive correlation between these two variables was simulated. Similarly for the negative correlation, between sign of skew and C j , only one of the two possible conditions was simulated. Table 2 summarizes the experimental conditions simulated for conditional normality and experimental conditions of Case A. After simulating all experimental conditions for the sample size of 20 in each group, a number of conditions were simulated for the sample size of 40. It became quite obvious that the trends found in simulations for samples of size 20 were, with few exceptions, occurring in similar fashion for identical conditions with the larger sample size of 40 in each group. Due to cost considerations, a limited number of experimental conditions involving a sample size of 40 were chosen. Case_B. Here again, there were eight possible patterns of positive and negative correlations among C, , Sj , and degree of skew. Data were simulated for only two groups. Therefore, given the correlations between two pairs of these variables, the third correlation was automatically determined, and there were again only four distinct patterns PAGE 41 35 Table 2 Summary of Conditions Investigated for Conditional Normality and Case A Size of the Conditional Variance 1 .3 2 1 .7 2 Sample Size Variance Shape of Correlation Ratio" Distribution 13 20,20 40,40 20,20 40,40 d 1:1 normal * * * * 2 : 1 normal * * * * + 1:1 + .25 * * * * + .75 * * * * 2:1 + .25 * * * + .75 * * * 1:1 T .25 * * + .75 * * 2:1 + .25 * * + .75 * * a. refers to the correlation between sign of skew and height of conditional mean function. b. refers to the conditional distribution. c. average conditional variance in the heteroscedastic cases Q. correlation not defined in these cases. PAGE 42 36 of correlations among the three variables. The relationship used to limit the number of simulations in Case A, Y* = -Y, was not applicable for the experimental conditions in Case B. As a result, all four patterns of correlations among Cj , S * , and degree of skew were simulated. Table 3 provides a summary of the experimental conditions simulated in Case B. Again, all experimental conditions were simulated with a sample size of 20 in each group and selected conditions with a sample size of 40 in each group were simulated. Results of these simulations were similar to those for the sample size of 20 and due to cost considerations the simulation of all experimental conditions for the sample size of 40 was not completed. Summary . A total of 64 experimental conditions involving nonnormal conditional distributions was possible for the factors described above. However, all possible experimental conditions were not simulated for all conditions where sample size was 40 in each group. For each of the conditions chosen from among the possible combinations, data were simulated both for C 2 C x equal to zero and for the four non-zero levels of C 2 C r When C 2 C Â± was equal to zero, the correlation between C. and direction or degree of skew was not defined. Therefore, for the non-normal conditional distributions, when C 2 C 1Â§ was equal to zero, the simulation really provided just two replicates of each condition. PAGE 43 37 Table 3 Summary of Conditions Investigated for Case B Size of Conditional Variance 3 1 .3 2 1 .7 2 , Sample Size Conditional Group Variance Skew Mean 20,20 40,40 20,20 40,40 1 + + * * k * 2 _ 1 + Â— * * * * 2 + 1 + + + * * * 2 _ 1 + + * * * 2 + + 1 + + * * * 2 + 1 + * * * 2 + + Â— Note: The conditions indicated were investigated for both combinations of skew, (.25, .50) and (.25, .75) a. refers to size of the average conditional variance for heteroscedastic cases. b. a blank for both groups indicated homoscedasticity . A + indicates the group with the larger conditional variance c. A + indicates the group with the more skewed distribution d. A + indicates the group with the larger conditional mean function. PAGE 44 CHAPTER IV RESULTS AND DISCUSSION Type I Error Probabilities One purpose of this study was to examine the estimated actual alpha levels of the parametric and rank analysis of covariance procedures, when those procedures were used to test the null hypothesis of equality of conditional means and the conditional distributions encountered differed in scale and/or shape. For each of the experimental conditions studied, observations were simulated from two populations that had equal conditional means . The simulations were replicated 1000 times for each experimental condition, and the resulting data were analyzed using the usual analysis of covariance procedure and the rank analysis of covariance procedure (McSweeney and Porter, 1971) described in an earlier chapter. For completeness, nominal alpha levels of .01 and .05 were used in each case. For each test and alpha level, the proportion of the 1000 replications resulting in a significant statistic provides an estimate of the actual alpha level. To identify those conditions for which a 38 PAGE 45 39 test did not preserve its level, a criterion interval of acÂ± 2joL( 1 oc) / 1000 was used for nominal alpha levels of .01 and .05. When #= .01 the resulting criterion interval was (.004, .016) and for PAGE 46 40 Table 4 Conditions Which Resulted in an Actual Probability of a Type I Error Outside the Criterion Level: Parametric ANCOVA Size of Nominal Alpha Conditional Sample Ratio of Skew Variance Size Variances .01 .05 + .25 1-.3 2 20 1 .020,.011 a +.25 1-.32 1-.32 1-.72 13 2 1-.3? 1-.7 2 20 2 .014, .011 + .25 40 1 .068, .051 + .75 .25/. 50 40 40 1 1 .018, .004 .054,. 065 .25/. 75 20 1 .017, .010 .25/. 75 40 1 .017, .011 .25/. 75 40 1 .003,. 009 Note: For all conditions identified in this table only one of the two replications of the condition resulted in a Type I error probability outside the criterion interval . a. estimated actual alpha levels. PAGE 47 41 both replications. These findings suggest that when the assumptions of conditional normality and/or homogeneity of conditional variances are violated, to the degree examined in this study, the parametric analysis of covariance procedure appears to have preserved its level. The results provided in Table 5 indicate that the rank analysis of covariance procedure yielded estimated actual alpha levels well above the upper criterion interval limit in both replications of each nominal alpha level for several of the experimental conditions simulated. When the conditional distributions had the same degree of skew (+ .75), but were skewed in opposite directions, and the size of the conditional variance was 1-.3 , aberrant alpha levels were observed for all combinations of sample size and ratio of variances . Unacceptable alpha levels were also observed for these conditional distributions (skew + .75) when the size of conditional variance was 1-.7 , sample size was 40, and the ratio of variances was 2. For all other experimental conditions identified in Table 5 , the estimated actual alpha levels were above the upper limit of the criterion interval for only one of the replicates and these alpha levels tended to be fairly close to the upper limit. It appears then, that the levels of the parametric and rank analysis of PAGE 48 42 Table 5 Conditions Which Resulted in an Actual Probability of a Type I Error Outside the Criterion Level: Rank ANCOVA Size of Nominal Alpha Conditional Sample Ratio of Skew Variance Size Variances .01 .05 Normal 1-7* 1-.3 2 , 1-.3 2 . 1-.7 Z 1-.3 2 , 1-.7? 1-.3; 40 1 .070Â° + .75 20 1 .022,.024 a .079, .078 a + .75 20 2 .023,.021 a .075, .069* + .75 20 2 .063, .066 b + .75 40 1 .022,.040 a .086, .095* + .75 40 1 .019,.016 b .048, .066 b + .75 40 2 .018,. 020? .074, .075 a .25/. 50 40 1 .017,. 011Â° .25/. 50 40 2 .017 ,.010Â° .25/. 75 1-.3? 40 1 .022,.015 b .25/. 75 40 2 .063, .064 b b. in both replications of this condition the actual probability of a Type I error was outside the criterion interval . in one of the two replications of this condition the actual probability of a Type I error was outside the criterion interval. there was only one replicate of this condition. PAGE 49 43 covariance procedures are preserved for the vast majority of conditions simulated in this study. The exceptions occurred when the rank analysis of covariance was employed with conditional distributions that were moderately skewed in opposite directions and the size of the conditional variance was large (l-.3^), or with conditional distributions that were moderately skewed in opposite directions and the groups (sample size 40) had unequal conditional variances and the size of the conditional variance was small (1-.7 ). These findings are comparable to those obtained in other research studies which examined empirically, other more extreme violations of the conditional normality and/or homogeneity of conditional variance assumptions (Abunnaja, 1983; Conover and Iman, 1980; Olejnik and Algina, 1983; Olejnik, Algina and Abdel-Fattah, 1984). Finally, while an investigation of the actual power of the rank analysis of covariance to detect differences in skew and/or variance was not a major objective of this study, the results given above would also suggest that for the levels of skew investigated, the rank procedure has little power to detect differentially skewed conditional distributions that do not differ in conditional means . PAGE 50 44 Power Differences A second, and perhaps most important purpose of the study, was to compare the statistical powers of the rank and parametric analysis of covariance procedures in two research situations . The first situation considered was one in which the researcher was interested in detecting differences in conditional means , and the conditional distributions encountered in the study differed in skew and/or variance across groups. In the second, the researcher was interested in detecting differences in conditional distributions and expected that the treatments would have resulted in conditional distributions that differed in mean and skew and/or variance. For each experimental condition investigated, observations were simulated from two populations which had conditional mean functions that differed by a factor of .20, .40, .60, or .80 marginal standard deviation units, when sample sizes were 20 and 20. When sample sizes were 40 and 40, the conditional mean functions differed by a factor of .15, .30, .45, or .60 marginal standard deviation units. The 1000 replications for each condition were analyzed using the parametric and rank analysis of covariance procedures and nominal alpha levels of .01 and .05. For each test and nominal alpha level, the proportion of the 1000 replications resulting in a significant statistic provided an estimate of the statistical power of the test. PAGE 51 45 The results are reported in a series of tables that, for clarity, have been further divided into two sections. In the first section, reported results pertain to experimental conditions where the assumption of conditional normality was the only assumption violated. In the second section, the reported results reflect the effects of a combination of conditional non-normality and between group heterogeneity of variance. Each table reports the power of the parametric analysis of covariance procedure (P ) and the power differences between the parametric and rank analysis of covariance procedures (P p -P ) for nominal <* = .05. The results for nominal <*= .01 were quite similar. For each condition in each table, four parametric power estimates and estimated power differences are provided, one for each of the four effect sizes investigated for that condition. Homogeneous Conditional Variances Case__A. Table 6 provides the parametric power estimates (P ) and the power differences (P -P ) for the normal conditional distributions and for the experimental conditions where the conditional distributions differed in skew and there was a positive correlation between the sign of skew and C, from (2). The most obvious characteristic of the results was that, with one exception, the estimated power difference were PAGE 52 46 Table 6 Power of the Parametric ANCOVA (P ) and Power Difference (Pp-P r ) between the Parametric and Rank ANCOVA Effect Conditional Distributions Normal + .25 a + . 75 a Sample Size P Size P P P n~ P PÂ„ P -P P P -P P r P P r P P r 20,20 .3 .20 104 b -001 129 021 115 067 .40 260 023 262 040 205 085 .60 492 027 500 061 456 131 .80 742 027 750 073 728 176 .7 .20 158 019 144 027 101 021 .40 393 037 407 056 358 090 .60 732 064 734 061 751 144 .80 929 029 927 029 957 053 40,40 .3 .15 102 001 100 011 095 050 .30 300 017 274 062 280 158 .45 565 022 544 086 550 232 .60 780 022 821 101 814 235 .7 .15 161 004 140 017 086 060 .30 450 029 466 056 310 123 .45 776 044 807 069 664 151 .60 960 024 967 008 913 050 a. all results refer to the conditions with a positive correlation between sign of skew and C. b. decimal points are omitted. J PAGE 53 47 positive indicating that the parametric analysis of covariance was a more powerful test for both the normal and the non-normal conditional distributions. Since, with the one exception, the power differences were positive, the effects of the factors manipulated in the study have little implication for the choice of a statistical test. Nevertheless, it may be of interest to note that for a particular sample size, size of conditional variance (here strength of covariatedependent variable correlation) , and effect size the power advantages favoring the parametric analysis of covariance actually increase as the conditional distributions become more non-normal. Consider the most extreme case, where sample size was 40 and /o =.3 the largest power difference for the normal conditional distributions was .02 while the largest power difference for the conditional distributions having degree of skew + .75 was .235. While less extreme changes occur for the other combinations of sample size and >o , substantial changes do occur for each combination. It is also of interest to note that, in general, for a particular combination of sample size, covariate-dependent variable correlation, and effect size the power of the parametric analysis of covariance is affected little by changes in the shape of the conditional distributions. The effect of shape of the conditional distributions on PAGE 54 48 power differences occurs because the power of the rank analysis of covariance tends to decrease as the conditional distributions become increasingly nonnormal . Table 7 gives the parametric power (P ) and the power differences (P p -P r ) for the experimental conditions where the conditional distributions differed in sign of skew and there was a negative correlation between sign of skew and Ci from (2) . The power differences were inconsequential when the conditional distributions had degrees of skew + .25. However, when conditional distributions had degrees of skew + .75 some reasonably important power differences favoring the rank analysis of covariance procedure occurred when sample size was 20 or 40 and/ 1 ? was .3 or when sample size was 40 and ^>was .7. The largest differences occurred when sample size was 40 and p was .3. Differences of this magnitude certainly have implication for the choice of a statistical procedure . In interpreting these power differences though, it must be recalled that when the rank analysis of covariance was used to test the equality of conditional means, the level of the test was not preserved when ^>was .3 and sample size was 20 or 40 (see Table 3). If the purpose of a particular analysis is to test the equality of conditional means the rank analysis PAGE 55 49 Table 7 Power of the Parametric ANCOVA (P ) and Power Difference (P -P r ) between the Parametric and Rank ANCOVA: Case A Sample Size 20,20 40,40 Effect Size .20 .40 .60 .80 .20 40 60 80 .15 30 45 60 .15 .30 .45 .60 Conditional Distributions a + .25' 133 214 491 715 137 395 705 966 101 277 545 768 183 486 873 948 P -P P r 002 -001 006 027 000 009 026 026 Â•019 027 001 017 Â•013 017 041 009 + .75" 120 281 492 682 149 408 693 903 143 297 547 767 137 421 740 936 P -P P r -049 -040 -043 -033 -031 008 Â•007 017 085 Â•119 102 069 -043 -046 -025 002 Note: All results refer to the conditions with a negative correlation between sign of skew and Ca. decimal points are omitted. J PAGE 56 50 procedure could not be recommended for these conditions. It would however be the preferred procedure when p is large (f> = .7) and sample size is large (n = 40) since the test did preserve its level in this condition and enjoys a small power advantage over the parametric procedure. For the remaining experimental conditions the choice between the two procedures appears to be relatively inconsequential. The levels of both tests were preserved when they were used to test the equality of conditional distributions. The rank analysis of covariance could therefore be recommended in this situation when p is small (*= .3) and sample size is either 20 or 40 or when /> is large (/> = .7) and sample size is 40. For all other experimental conditions, the choice between statistical procedures would again be relatively inconsequential. The results do suggest, however, that a choice may not be inconsequential with larger sample sizes. When conditional distributions were skewed + .25 and the covariate-dependent variable correlation was small ( n = .3) the parametric analysis of covariance enjoyed power advantages for a sample size of 20. When sample size was increased to 40, though, the rank analysis of covariance procedure enjoyed power advantages of approximately the same magnitude. When the conditional distributions were skewed + .75 and the covariate-dependent variable PAGE 57 51 correlation was weak ( p .3) the rank analysis of covariance enjoyed power advantages for both sample sizes, but those power advantages increased with the increase in sample size. Taken together these results would suggest that when the covariate-dependent variable correlation is weak and sample sizes are large ( n>40) the rank analysis of covariance procedure may be preferred even with conditional distributions slightly skewed in opposite directions. Case_B. Table 8 gives the parametric power (Pp) and the power differences (P p -P r ) for those experimental conditions where conditional distributions differed in degree of skew and there was a positive correlation between degree of skew and C from (2). Here, all power differences are positive suggesting that the parametric analysis of covariance procedure is more powerful than the rank analysis of covariance for all combinations of sample size, strength of covariate-dependent variable correlation, and magnitude of difference in degree of skew. Thus, when conditional distributions differ in degree of skew and there is a positive correlation between degree of skew and C from (2) , the parametric analysis of covariance procedure would be preferred. PAGE 58 Table 8 52 Power of the Parametric ANCOVA (P ) and Power Differences ( p pp r ) between the Parametric and Rank ANCOVA: Case B Conditional Distributions Â•25, .50 .25, .75 Sample Size 20,20 .3 40,40 Effect Size .20 .40 .60 .80 20 40 60 80 15 30 45 60 .15 .30 .45 .60 088 242 495 753 124 406 727 924 097 268 543 787 154 461 803 968 P r 007 031 040 058 006 050 066 037 007 040 057 052 026 059 041 021 a. b. 106 245 490 715 140 415 723 932 109 298 542 822 149 449 791 966 p ^p P r 015 029 040 048 014 066 062 036 017 076 073 081 021 068 069 039 all results refer to the conditions with a positive correlation between C* and degree of skew, decimal points are omitted. PAGE 59 53 Table 9 gives the parametric powers (P ) and power differences (PÂ„-P ) for the experimental conditions where the conditional distributions differed in degree of skew and there was a negative correlation between degree of skew and Cj from (2). In general, the power differences are of little consequence. However, the parametric analysis of covariance enjoyed a small power advantage for both levels of difference in skew when the covariate-dependent variable correlation was large (/3=.7) and sample size was 20. The rank analysis of covariance enjoyed a power advantage of about the same magnitude when the conditional distributions had degrees of skew .25 and .75, and the covariate-dependent variable correlation was weak (p=.2>) and sample size was 40. It is also of interest to note that for a given level of difference in degree of skewness and strength of covariate-dependent variable correlation, an increase in sample size had an obvious effect on power difference. As sample size increased, any power differences in favor of the parametric analysis of covariance tend to decline and the power differences in favor of the rank analysis of covariance tend to increase. This would suggest that with even larger sample sizes, the rank analysis of covariance may enjoy practically important power advantages for a number of experimental conditions involving various combinations of strength of covariate-dependent variable correlation and difference in degree of skew. The work of PAGE 60 54 Table 9 Power of the Parametric ANCOVA (P ) and Power Differences ( p pp r ) between the Parametric and Rank ANCOVA: Case B Sample Size 20,20 .7 40,40 .3 Effect Size .20 .40 .60 .80 20 40 60 80 .15 30 45 60 .15 .30 .45 .60 Conditional Distributions Â•25, .50 .25, .75 P -P P r 109 -003 277 025 486 010 710 006 136 -002 415 051 696 040 908 035 103 -008 275 -013 524 000 774 004 157 005 440 009 798 031 957 005 P -P P r 107 -018 260 -013 475 -024 734 -012 146 005 382 061 733 047 915 033 122 -026 289 -059 542 -042 755 -036 145 -014 450 011 788 020 955 007 all results refer to the conditions with a negative correlation between C. and degree of skew. b. decimal points are omitted. PAGE 61 55 Olejnik and Algina (1984) would suggest, however, that the greatest power increases might be expected in the middle of the power curves where the parametric power ranged from .4 to .6. Heterogeneous Conditional Variances A number of experimental conditions were simulated in this study so that the combined effects of conditional non-normality and between group heterogeneity of variance could be examined. The results obtained from the simulation of these experimental conditions were quite similar to those obtained for experimental conditions where conditional non-normality was the only violation considered, and detailed interpretation of the results obtained in the heterogeneous case would therefore be redundant. For this reason, a very general discussion of the results is given. Case A . Table 10 provides the parametric power estimates (P ) and the power differences (P_-P r ) for the normal conditional distributions and for the experimental conditions where the conditional distributions differed in sign of skew and there was a positive correlation between sign of skew and C, from equation (2) . The power differences are all positive suggesting that the parametric analysis of covariance was again a more powerful test for both the normal and non-normal conditional distributions. Also, for a particular sample PAGE 62 56 Table 10 Power of the Parametric ANCOVA (P ) and Power Difference (P -P ) between the ParametriE and Rank ANCOVA Size of Conditional Distributions Normal + .25 + .75 Sample Conditional Effect Size Variance Size P n p r,p P P -P P P -P P P r P P r P P r 20,20 1-.3 2 .20 095 a 005 078 001 073 018 .40 236 004 214 031 216 072 .60 460 027 459 060 480 135 .80 716 054 710 054 737 136 20,20 1-.7 2 .20 144 018 148 019 110 004 .40 370 034 419 062 391 097 .60 724 051 754 080 734 111 .80 931 035 939 039 952 050 40,40 1-.7 2 .15 138 022 137 021 133 040 .30 475 048 433 075 424 121 .45 798 030 779 079 786 132 .60 951 015 956 029 973 060 Note: All results refer to heterogeneous variance conditions where there was a positive correlation between sign of skew and C. a. decimal points are omitted. PAGE 63 57 size, size of conditional variance, and effect size the power advantages favoring the parametric analysis of covariance increase as the conditional distributions become more non-normal . Table 11 gives the parametric powers (P ) and the P power differences (P p -P r ) for the experimental conditions where the conditional distributions differed in sign of skew and there was a negative correlation between sign of skew and Cj from (2). Again, power differences appear to have been inconsequential when the conditional distributions had degrees of skew + .25. When degree of skew was + .75, however, some reasonably important power differences favoring the rank analysis of covariance procedure were observed. As with the homogeneous variance conditions, the greatest power advantages in favor of the rank analysis procedure occurred when the size of the conditional variance was large (1-.32). When interpreting these power differences it must be recalled that when the rank analysis of covariance is used to test the equality of conditional means, aberrant alpha levels were observed when the size of the conditional variance was large (1-.3 2 ) and sample size was 20 or 40, or when the size of the conditional variance was small (1-.7 ) and sample size was 40 (see Table 5). If the purpose of analysis is to test the equality of conditional means, the rank analysis of covariance could not be PAGE 64 58 Table 11 Power of the Parametric ANCOVA (P ) and Power Difference (Pp-P r ) between the Parametric an9 Rank ANCOVA: Case A Conditional Distributions + .25 + .75 Size of Sample Conditional Effect Size Variance Size 20,20 20,20 40,40 40,40 1-.3' 1-.7' 1-.32 1-.72 .20 .40 .60 .80 .20 .40 .60 .80 .15 .30 .45 .60 .15 .30 .45 .60 P r 098 a 000 236 002 449 008 701 012 151 -007 412 016 728 036 930 051 177 017 463 001 780 033 954 016 Note: P -P P r 119 -039 301 -044 504 -044 712 -007 175 -025 410 -010 690 019 899 041 174 -032 472 -041 772 -010 935 004 a. All results refer to the heterogeneous variance conditions with a negative correlation between sign of skew and C decimal points are omitted. j PAGE 65 59 recommended for these conditions . For the remaining experimental conditions a choice between the two procedures would be inconsequential. When testing the equality of conditional distributions, the levels of both tests were preserved, thus the rank analysis of covariance may be a more powerful procedure when the size of the conditional variance is large (l-.3^) and the conditional distributions are moderately non-normal (skew + .75) . Case B . In this case, there were eight possible patterns of positive and/or negative correlations among Cj , Sj , and degree of skew. Since data were simulated for just two groups, once the correlations between two pairs of variables was specified, the third correlation was automatically determined. This meant that there were four distinct patterns of correlation among the three variables. The relationship used to limit the number of simulations in Case A, Y* = -Y, was not applicable for experimental conditions in Case B. As a result all four patterns of correlations among C, S* , and degree of skew were simulated. In Tables 12 through 15 the parametric power estimates (P ) and estimated power differences (P p -P r ) are reported for the experimental conditions where the conditional distributions were differentially skewed, in the same direction. Each table provides results for experimental conditions for one of PAGE 66 60 Table 12 Power of the Parametric ANCOVA (P ) and Power Difference (Pp-P ) between the Parametric ana Rank ANCOVA: Case B Conditional Distributions .25, .50 .25, .75 Size of Sample Conditional Effect Size Variance Size 20,20 1-.3 20,20 1-.7 2 40,40 1-.3 2 40,40 1-.7 2 .20 .40 .60 .80 ,20 40 60 80 .15 .30 45 60 .15 ,30 45 60 p P -P P P -P p P r P P r 097 a 012 091 025 229 032 215 050 483 075 434 103 724 071 700 102 134 016 128 020 384 064 407 076 727 073 754 075 939 048 945 042 146 033 443 086 795 082 959 028 129 039 421 098 785 095 958 027 Note: All results refer to heterogeneous variance conditions with a positive correlation between degree of skew and C. and a positive correlation between degree of skew and S . . a. decimal joints are omitted. PAGE 67 61 Table 13 Power of the Parametric ANCOVA (P ) and Power Difference (Pp-P ) between the Parametric and Rank ANCOVA: Case B Size of Sample Conditional Effect Size Variance Size 20,20 20,20 40,40 1-.3' 1-.7' 1-.3.20 .40 .60 .80 20 40 60 80 15 30 45 60 Conditional Distributions .25, .50 P -P P r 104a 008 250 016 506 038 722 037 158 009 406 030 731 050 917 023 25, .75 P -P P r 099 002 246 001 502 027 744 049 150 -007 396 008 706 018 925 021 40,40 Note l-.7< .15 30 45 60 157 010 464 009 795 034 957 014 170 016 450 041 780 031 953 041 All results refer to the heterogeneous variance conditions with a positive correlation between degree of skew and C. and a negative correlation between degree of skew and S.J decimal points are omitted. J PAGE 68 62 Table 14 Power of the Parametric ANCOVA (P ) and Power Difference (P -P ) between the Parametric and Rank ANCOVA: Case B P r Conditional Distributions Size of .25, .50 25, .75 Sample Conditional Effect Size Variance Size P p r,p P P -PÂ„ P P r P P r 20,20 1-.3 2 .20 112 a -007 110 -034 .40 235 -021 268 -044 .60 472 015 494 -021 .80 708 011 711 -005 20,20 1-.7 2 .20 141 001 120 -007 .40 367 030 375 010 .60 696 041 674 022 .80 891 020 910 035 40,40 1-.3 2 .15 .30 .45 .60 40,40 1-.7 2 .15 166 -014 149 026 .30 452 006 464 -015 .45 773 027 768 -007 .60 955 026 945 011 Note: All results refer to heterogeneous variance conditions with a negative correlation between degree of skew and C. and a positive correlation between degree of skew and S . . a. decimal points are omitted. PAGE 69 63 Table 15 Power of the Parametric ANCOVA (P ) and Power Difference ( p p _p r ) between the Parametric anB Rank ANCOVA: Case B Size of Conditional Distributions Â• 25, 50 .25, .75 Sample Conditional Effect Size Variance Size P P V 1 r P P P -P r P r 20,20 1-.3 2 .20 105 a 013 089 -015 .40 264 025 239 022 .60 510 054 478 041 .80 721 040 719 030 20,20 1-.7 2 .20 142 014 129 011 .40 398 023 411 042 .60 727 055 715 044 .80 930 052 926 048 40,40 1-.3 2 .15 .30 .45 .60 40,40 1-.72 .15 148 012 152 011 .30 441 022 440 042 .45 804 063 760 050 .60 964 018 948 023 Note: All results refer to heterogeneous variance conditions with a negative correlation between degree of skew and ndS a negatlve cor relation between degree of skew a. decimal joints are omitted. PAGE 70 64 the four possible patterns among C^ , Si , and degree of skew. Careful review of the entries in these tables would indicate that, with few exceptions, the power differences are positive suggesting that the parametric analysis of covariance procedure was a more powerful test. The greatest power differences favoring the parametric procedure were obtained for experimental conditions where there was a positive correlation between degree of skew and Cand a positive correlation between degree of skew and Sj (Table 12) . The rank analysis of covariance enjoyed small power advantages in experimental conditions where the correlation between degree of skew and Cj was negative, the correlation between degree of skew and S^ , was positive, the conditional distributions had degrees of skewness .25, .75, the size of the conditional variance was large (1-.3 ) and the sample size was 20. (See Table 14) For all other experimental conditions , the choice of a statistical procedure would be inconsequential. However, the preponderance of power differences favored the parametric procedure suggesting that in the heterogeneous case, when conditional distributions are differentially skewed, a general use of the parametric procedure is indicated. PAGE 71 CHAPTER V CONCLUSIONS An important consideration for the researcher who must choose between a parametric or nonparametric statistical test is the effect that violations of assumptions, separately and in combination, have on type I errors and statistical power for the alternative procedures. Specific information of this kind for a variety of combinations and degrees of violations of assumptions could be most useful to the researcher. Armed with this information, astute decisions regarding the appropriateness of a statistical procedure may be made. In the present study a limited number of discrepancies between expected and estimated actual alpha levels were observed for a large number of experimental conditions where the assumptions of conditional normality and/ or between group homogeneity of variance were violated to varying degrees. Those aberrant alpha levels have been reported in Tables 4 and 5. Further, comparisons of the statistical powers of the parametric and rank analysis of covariance procedures were made for those same experimental conditions. The results of those 65 PAGE 72 66 comparisons have been highlighted in Tables 6 through 15. While detailed discussions followed each of the Tables 4 through 15, certain conclusions follow. These conclusions may provide guidelines for researchers who must choose between the parametric and rank analysis of covariance procedures. Conclusion 1 . The level of the parametric analysis of covariance is preserved for moderate violations of the conditional normality and/or between group homogeneity of variance assumptions. As evidenced in Table 4, when sample sizes are equal, conditional non-normality and/or between group heterogeneity of variance produced few discrepancies between expected alpha levels and estimated actual alpha levels. Further, even when aberrant alpha levels were observed for a particular experimental condition those aberrant levels were not replicated. Thus, when faced with only moderate violations of the conditional normality and/or between group homogeneity of variance assumptions the researcher need not dispose of the parametric analysis of covariance in favor of some alternative distribution free procedure. Conclusion 2 . The level of the rank analysis of covariance is, with few exceptions, preserved when conditional distributions are moderately non-normal and conditional variances across groups are unequal (ie ratio of 2). The exceptions were identified in Table 5. In PAGE 73 67 general though they indicate that the rank analysis of covariance may be quite sensitive to moderate differences in direction of skew when the size of the conditional variance is large. The rank analysis of covariance may therefore be an inappropriate test of a given hypothesis when conditional distributions differ in this way. Conclusion 3 . When testing the null hypothesis of equality of conditional means, the parametric analysis of covariance is the procedure of choice. Tables 6 through 15 indicate that the parametric analysis of covariance, by and large, enjoys power advantages for most experimental conditions examined in this study. When the rank analysis of covariance had any power advantages they often occurred for experimental conditions where the level of the rank procedure was not preserved for this particular hypothesis test. In short very little power will be lost and possibly much power gained by the more general use of the parametric procedure to test the equality of conditional means for the degrees of violations examined in this study. Conclusion 4 . When testing the null hypothesis of equality of conditional distributions, and there is a positive correlation between sign or degree of skew and height of the conditional mean function, the parametric analysis of covariance is consistently more powerful and PAGE 74 68 is the preferred procedure. The power differences reported in Tables 6, 8, 10, 12, and 13 (conditions where there was a positive correlation between sign or degree of skew and C . ) are, with very few exceptions, positive indicating greater power for the parametric analysis of covariance. This would suggest that there would be little reason to use the rank analysis procedure to test the hypothesis of equality of conditional means or equality of conditional distributions. In fact, a large number of the power differences found in these tables are quite large suggesting practically important power advantages in favor of the parametric analysis of covariance . Conclusion 5 . When testing the null hypothesis of equality of conditional distributions, and there is a negative correlation between sign or degree of skew and height of conditional mean function, the choice of a statistical procedure is often inconsequential with the parametric procedure enjoying power advantages for some conditions and the rank procedure enjoying comparable power advantages for other dissimilar conditions. Careful review of the power differences in Tables 7, 9, 11, 14 and 15 does however lead one to the conclusion that the rank analysis of covariance enjoys power advantage for a limited number of experimental conditions. Those conditions often involved conditional distributions that were moderately PAGE 75 69 non-normal (+ .75 or .25, .75) having a large conditional variance (l-.3^). The parametric analysis of covariance, on the other hand, enjoyed power advantages for a greater number of experimental conditions where the conditional distributions were mildly to moderately non-normal and the size of the conditional variance was small (1-.7 Z ). This would suggest that if a reasonable covariate has been selected (ie. one resulting in a large covariate-dependent variable correlation) the parametric procedure may be preferred even for those conditions where there is a negative correlation between sign or degree of skew and height of the conditional mean function Limitations . There were obvious limitations in the present study that restrict its generalizability . First, the present study only considered slight to moderate violations of the conditional normality and between group homogeneity of variance assumptions. More extreme violations of the assumptions could result in very different conclusions regarding the choice of a statistical procedure. However, the aim of the present study was to consider degrees of violations that would be likely in actual practice. Second, the inability (due to cost considerations) to simulate all of the experimental conditions for a sample size of 40 was seen as a limitation of the study, although not a severe one. The experimental conditions simulated PAGE 76 70 suggest certain patterns in the power differences that are rather obvious. While the missing observations therefore appear to be obvious there is no concrete evidence to support this claim. Finally, only the rank transform alternative to the analysis of covariance was considered in this study. It is true that the rank transform procedure and Quade's and Puri and Sen's procedures have exhibited similar Type I error rates and power in other simulations However, that finding does not rule out the possibility that these three procedures may yield different results with the kind of data generated in this study. It seems then that a comparison of the nonparametric procedures applied to data of the kind generated here, would be most useful. Given these limitations, the results of this study should provide useful information for the researcher who must choose between the parametric and rank analysis of covariance procedures in the face of modest degrees of conditional non-normality and between group heterogeneity of variances. PAGE 77 BIBLIOGRAPHY Abunna j a , S . S . 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Cambridge, Massachusetts, Harvard University Press, 1962 Shields, J.L. An empirical investigation of the effect of heteroscedasticity and heterogeneity of variance on the analysis of covariance and the Johns on -Neyman technique. Technical paper no. 292, U.S. Army Research Institute for the Behavioral and Social Sciences, Alexandria, Virginia, 1978. Shirley, E.A. A distribution-free method for analysis of covariance based on ranked data. Journa l of Applied Statistics , 1981, 30, 158-162T Thomson, D.S. A Monte Carlo study: Robustness of analysis of covariance to violations of selected assumptions (Doctoral Dissertation, University of Illinois, 1980) Dissertation Abstracts International . 1981, 41, 4176-B PAGE 81 75 Wildemaim, C.E. The effect of the shape and length of the covariate score scale upon the F-test of the analysis of covariance (Doctoral Dissertation, University of Pittsburgh, 1974) Dissertation Abstracts International , 1974, 35, 2067-A. Winer , B.J. Statistical Principles in Experimental Design . New York: McGraw-Hill, 1962. Zikri, L. The robustness of the analysis of covariance to the violation of the assumptions of normality and homogeneity of variance . Paper presented at the meeting of the American Educational Research Association, Montreal, Canada, 1983. PAGE 82 76 BIOGRAPHICAL SKETCH Samuel L. Seaman was born 12 November, 1954, in Brookhaven, Pennsylvania. He graduated from DeLand Senior High School in June, 1972. In May, 1976, he received the Bachelor of Arts degree from Stetson University. He taught school for three years and in 1979 returned to Stetson University to do graduate work, where he received the Master of Education degree in May, 1980. In August, 1981 he began work toward the Ph.D. degree at the University of Florida. PAGE 83 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. James J. Algina, Chairman Associate Professor of Foundations of Education I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Gordon G. Bechtel Professor of Marketing I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. /// > Linda M. Crocker Associate Professor of Foundations of Education I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Stephen F. Olejnik Associate Professor of Foundations of Education PAGE 84 This dissertation was submitted to the Graduate Faculty of the Department of Foundations of Education in the College of Education and to the Graduate School, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1984 w,^-v i,j Â• . ,-.:..Chairman, Foundations' 1 of Education Dean for Graduate Studies and Research |