MetaAnalysis in Higher Education: An Illustrative Example Using Hierarchical Linear Modeling
Abstract
The purpose of this article is to provide higher education researchers with an illustrative example of metaanalysis utilizing hierarchical linear modeling (HLM). This article demonstrates the stepbystep process of metaanalysis using a recentlypublished study examining the effects of curricular and cocurricular diversity activities on racial bias in college students as an example (Denson, Rev Educ Res 79:805–838, 2009). The authors present an overview of the metaanalytic approach and describe a metaanalysis from beginning to end. The example includes: problem specification; research questions; study retrieval and selection; coding procedure; calculating effect sizes; visual displays and summary statistics; conducting HLM analyses; and sensitivity analyses. The authors also offer guidelines and recommendations for improving the conduct and reporting of research which in turn can provide the information necessary for future and more comprehensive metaanalytic reviews.
Keywords
Metaanalysis Effect size Randomeffects model Hierarchical linear modeling Moderator analysisWhile metaanalytic techniques have been available for over a century (see Cooper and Hedges (1994) for a history of metaanalysis), metaanalysis as a method of inquiry for integrating quantitative results from a stream of research began to become popular approximately 30 years ago (Glass 2000). While conducting a metaanalysis can be very time consuming when done properly, advances in computing and statistical software have facilitated conducting literature searches and applying metaanalytic procedures. The purpose of this article is to provide higher education researchers with a pedagogical example of a metaanalysis utilizing hierarchical linear modeling (HLM) to encourage more widespread use of metaanalytic methods. In this article, we begin by first providing an overview of the metaanalytic approach. We then provide an example of metaanalysis using Denson’s Review of Educational Research (2009) study of the effects of curricular and cocurricular diversity activities on racial bias in college students. Finally, we provide guidelines and recommendations for improving the conduct and reporting of future research.
Overview of the MetaAnalytic Approach
Put simply, metaanalysis is a statistical technique of combining the findings of a set of studies that address common research hypotheses (Cooper and Hedges 1994; Lipsey and Wilson 2001). As Glass (1976) defines it, “metaanalysis refers to the analysis of analyses…the statistical analysis of a large collection of analyses results from individual studies for the purpose of integrating the findings” (p. 3). Conducting a metaanalysis is much like conducting a primary research study. In a metaanalysis, instead of students or participants being the unit of analysis, the primary studies themselves become the unit of analysis. In metaanalysis, the researcher collects research studies from a particular domain (e.g., studies of the effects of curricular diversity activities), codes them, and synthesizes results from them using statistical methods analogous to those used in primary data analysis (e.g., techniques analogous to regression). As will be discussed in detail below, the key quantitative results of each study (e.g., comparisons of those students who participate in a diversity program of interest versus those who do not) are converted to effect sizes. The effect sizes for a sample of studies become the dependent variable, and various study characteristics (e.g., variables that capture the demographic characteristics of each study’s sample; variables that capture key features of each study’s program) become the independent variables. In this way, a metaanalysis provides an integrated review of the research base that enables us to calculate an overall summary of the effects of a program or policy of interest, and investigate various hypotheses concerning why the effects may be larger in some studies and smaller in others; that is, we can attempt to identify key factors that might magnify or dampen the effects of a program of interest.
Many extremely thoughtful and thorough qualitative research syntheses have been undertaken in education (e.g., Feldman and Newcomb 1969; Pascarella and Terenzini 1991, 2005). Metaanalysis complements qualitative literature reviews by integrating the findings in a systematic quantitative way, which we will discuss in detail below. Some examples of metaanalyses in education include: coaching effects on SAT scores (Messick and Jungeblut 1981), teacher expectancy effects on student IQ (Raudenbush 1984), school desegregation effects on AfricanAmerican students (Wells and Crain 1994), and effects of school reform programs on achievement (Borman et al. 2003).
 (1)
What is the magnitude of the overall effect size for a program or practice of interest?
 (2)
To what extent do the effect sizes of the studies in a sample vary around the overall effect size estimate?
 (3)
What are the reasons for the observed differences in effect sizes across studies? The presence of heterogeneity provides an opportunity to ask additional questions, and to investigate more closely the reasons for the observed differences in effect size across studies. Such investigations have the potential to generate further insight into what conditions and for whom a program or practice of interest may be particularly beneficial.
An Illustrative MetaAnalysis Example
Problem Specification
In the last 50 years, the rapidly changing demographics of the United States have seen a parallel shift in the demographic makeup of our college campuses. Naturally, this change has been met with some resistance and racial tension. As a result, institutions have implemented diversityrelated initiatives designed to promote positive intergroup relations. While there certainly have been many qualitative literature reviews on this topic (e.g., Engberg 2004), there had yet to be a quantitative review that summarized and integrated the findings of this topic thus far. Denson’s (2009) study was the first to address this research gap.
Research Questions
 (1)
What is the magnitude of the overall effect size of participation in curricular and cocurricular diversity activities on racial bias? (overall effect)
 (2)
Is there variation in the effect of curricular and cocurricular diversity activities on racial bias? (heterogeneity in effect sizes)
 (3)
Which types of programs are most effective? Which types of studies show the most effective programs? (factors that amplify or diminish the magnitude of effect sizes)
Study Retrieval and Selection

KEYWORDS = (diversity or ethnic studies or women studies)

AND

KEYWORDS = (bias or prejudice or stereotypes or discrimination)

AND

DESCRIPTORS = (higher education or college students)
Characteristics of the Sample of Studies (N = 27) (Copyright 2009 American Educational Research Association. Used by permission)
Category  Number of studies  Percentage 

Source of the studies  
Journal article  17  63 
Book or book chapter  3  11 
Conference paper  3  11 
Doctoral dissertation  3  11 
Article under review  1  4 
Year of publication  
1977–1999  15  56 
2000–2006  12  44 
Single/multiple institution  
Single institution  13  48 
Multiple institution  13  48 
Both  1  4 
Type of intervention  
Diversity course  8  30 
Ethnic studies course  10  37 
Women’s studies course  6  22 
Diversity workshop/training  15  56 
Peer facilitated training  5  19 
Combination  3  11 
Coding Procedure
The next step is to code these studies for their study characteristics and effect sizes which will become the independent variables and dependent variable, respectively (Lipsey and Wilson 2001). The study characteristics are possible moderators which may help explain the variation seen in effect sizes across studies. It is likely that there are systematic differences across studies that may account for differences in effect sizes. Effect size estimates are standardized measures of an effect and comprise the key outcome measure in metaanalysis and enable the synthesizing and integrating of findings.
Study Characteristics
The independent variables selected and coded for this metaanalysis were included as possible covariates, and were categorized into three types: study characteristics, student characteristics, and institutional characteristics. The study characteristics consisted of three subgroups: study source characteristics (a through c), diversity initiative characteristics (d through i), and methodological characteristics (j through q).
The study source characteristics included: (a) author identification, (b) year of publication, and (c) type of publication (journal article, book, book chapter, conference paper, doctoral dissertation and other unpublished work. The diversity initiative characteristics included: (d) type of curricular or cocurricular diversity activity (required diversity course, nonrequired diversity course, ethnic studies, women’s studies, diversity workshop or training intervention, peerfacilitated intervention), (e) pedagogical focus (enlightenment only, enlightenment and contact), (f) intensity of crossracial interaction (random, builtin), (g) duration of intervention (in weeks), (h) intensity of intervention (in hours/week), and (i) year of intervention. The methodological characteristics included: (j) outcome measures (k) reliability of outcome, (l) measurement technology of outcome (homegrown, wellestablished), (m) target group (general, specific), (n) quality of covariate adjustment (none, one/pretest, additional/multiple), (o) sample size, (p) sample assignment (not random, random), and (q) sample matching (no, yes).
The student characteristics included: (a) age, (b) gender, and (c) ethnicity. Finally, the institutional characteristics included: (a) number of institutions in study (single, multiple), (b) institution name, (c) institutional diversity requirement (no, yes), (d) type of institution (public, private), (e) size of institution (total number of fulltime undergraduates), (f) structural diversity (percentage of students of color), and (g) region of country (Western, Southern, Eastern, Midwest). All of the primary studies were coded according to these variables.
In addition, because coding might be subject to human error, two coders were involved in the coding process and interrater reliability indices were calculated accordingly. Specifically, Cohen’s (1960) kappa was computed for the categorical variables and Cronbach’s (1951) alpha was computed for the continuous variables. The interrater reliability indices were high, with the Cohen’s kappa statistics ranging from 0.80 to 1.00 and the Cronbach’s alpha coefficients ranging from 0.95 to 0.99.
Calculating Effect Sizes
The dependent variable was effect size and was estimated as a group contrast in this metaanalysis. A group contrast compares an outcome variable across two or more groups of respondents. For example, the racial bias measures can be compared between students who had taken a required diversity course versus those who had not. The group contrasts were represented by the standardized mean difference because the operationalization of racial bias differed to some extent across the studies, and it is thus necessary to standardize the group contrasts so that the values can be meaningfully compared across the sample of studies.
The studies in the sample for this study were essentially quasiexperimental in that students selected into the various diversity programs, and in nearly all cases the authors of these studies attempted to statistically adjust for any preprogram differences in initial bias. When possible, adjusted effect sizes were calculated from inferential statistics (e.g., ANCOVA, multiple regression) reported in these articles (for details concerning computing adjusted effect sizes and their error variances, see Borenstein 2009). In these cases, the effect sizes can be viewed as being adjusted for differences in covariates (e.g., a pretest). When insufficient information was reported such that it was not possible to obtain the adjusted effect size, the unadjusted effect size was calculated from the reported descriptive statistics (e.g., means, standard deviations, correlations) using standard metaanalytic procedures (Hedges and Olkin 1985; Lipsey and Wilson 2001). Thus, whether or not a study’s effect size is adjusted or not is coded, and in our analyses we will examine the extent to which the pattern of results tend to differ for the set of studies with adjusted effect sizes and those with unadjusted effect sizes.
Summary of effect sizes by study
Study  Pedagogical approach  Type of intervention  Type of sample  Type of effect size  ES estimates (g)  SE of estimates (SE(g)) 

Katz and Ivey (1977)  Pedagogical approach  Diversity workshop and training intervention  Whites  Unadjusted  2.17  0.52 
Lopez et al. (1998) [1]  Pedagogical approach and CRI  Nonrequired diversity course and peer facilitated intervention  Mixed  Adjusted  0.87  0.16 
Muthuswamy et al. (2006) [1]  Pedagogical approach and CRI  Peerfacilitated intervention  Mixed  Unadjusted  0.86  0.22 
Pedagogical approach  Diversity workshop and training intervention  Whites  Unadjusted  0.83  0.03  
Antony (1993) [3]  Pedagogical approach  Diversity workshop and training intervention  Mixed  Unadjusted  0.81  0.02 
Taylor (1994) [1]  Pedagogical approach  Combination  Whites  Unadjusted  0.67  0.11 
MacPhee et al. (1994) [2]  Pedagogical approach  Nonrequired diversity course  Mixed  Unadjusted  0.64  0.09 
Antony (1993) [1]  Pedagogical approach  Ethnic studies course  Mixed  Unadjusted  0.61  0.02 
Pedagogical approach  Ethnic studies course  Whites  Unadjusted  0.59  0.02  
Hogan and Mallott (2005) [2]  Pedagogical approach  Required diversity course  Mixed  Unadjusted  0.53  0.16 
Pedagogical approach  Diversity workshop and training intervention  Students of color  Unadjusted  0.52  0.10  
Antony (1993) [2]  Pedagogical approach  Women’s studies course  Mixed  Unadjusted  0.48  0.02 
Gurin et al. (2004) [1]  Pedagogical approach and CRI  Nonrequired diversity course and peer facilitated intervention  Mixed  Adjusted  0.48  0.18 
Pedagogical approach  Ethnic studies course  Students of color  Unadjusted  0.46  0.10  
Gurin et al. (2002) [1]  Pedagogical approach  Ethnic studies course  Whites  Unadjusted  0.44  0.02 
Vogelgesang (2000) [3]  Pedagogical approach  Diversity workshop and training intervention  Mixed  Unadjusted  0.43  0.02 
Vogelgesang (2000) [6]  Pedagogical approach  Diversity workshop and training intervention  Students of color  Unadjusted  0.42  0.08 
Vogelgesang (2000) [9]  Pedagogical approach  Diversity workshop and training intervention  Whites  Unadjusted  0.41  0.02 
Vogelgesang (2000) [5]  Pedagogical approach  Women’s studies course  Students of color  Unadjusted  0.40  0.08 
Stake and Hoffman (2001) [3]  Pedagogical approach  Women’s studies course  Mixed  Adjusted  0.39  0.09 
Chang (2002)  Pedagogical approach  Required diversity course  Mixed  Adjusted  0.38  0.15 
Stake and Hoffman (2001) [2]  Pedagogical approach  Women’s studies course  Mixed  Adjusted  0.37  0.09 
Vogelgesang (2000) [8]  Pedagogical approach  Women’s studies course  Whites  Unadjusted  0.36  0.02 
Vogelgesang (2000) [2]  Pedagogical approach  Women’s studies course  Mixed  Unadjusted  0.35  0.02 
Gurin et al. (2002) [2]  Pedagogical approach  Ethnic studies course  Students of color  Unadjusted  0.33  0.12 
Vogelgesang (2000) [1]  Pedagogical approach  Ethnic studies course  Mixed  Unadjusted  0.31  0.01 
Antonio (2001)  Pedagogical approach  Diversity workshop and training intervention  Mixed  Adjusted  0.30  0.04 
Vogelgesang (2000) [7]  Pedagogical approach  Ethnic studies course  Whites  Unadjusted  0.30  0.02 
Vogelgesang (2000) [4]  Pedagogical approach  Ethnic studies course  Students of color  Unadjusted  0.25  0.08 
HendersonKing and Kaleta (2000) [2]  Pedagogical approach  Required diversity course  Mixed  Unadjusted  0.15  0.11 
Table 2 displays the 30 effect size estimates and their standard errors. As can be seen the effect size estimates range in value from 0.15 to 2.17, and the standard errors of the estimates range from 0.01 to 0.52. To help understand the meaning of these quantities, consider, for example, the results for the Lopez et al. (1998) study. The effect size estimate for this study indicates that the outcome scores for the students in the treatment group were on average approximately 0.87 standard deviations higher than the outcome scores for the students in the control group (with higher scores indicating greater racial understanding or less bias). According to Cohen (1988), effect size estimates around 0.2–0.3 are considered to be small, around 0.5 are moderate, and estimates greater than 0.8 are large.
Just as a treatment effect estimate provides us with an estimate of the true effect of a program or treatment, an effect size estimate (g _{ j }) can be viewed as providing an estimate of the true effect size for a study (δ _{ j }).^{1} The standard error gives us a sense of how precise the effect size estimate is, and enables us to construct a confidence interval, which helps convey how large or small the true effect might be. Thus, for the Lopez et al. study, adding and subtracting approximately 2 standard errors (2 × 0.16) to the effect size estimate (0.87), we obtain a 95% interval whose lower boundary is 0.56 and whose upper boundary is 1.18. Based on this interval, the notion that the true effect might be 0 is clearly not plausible; as can be seen in Fig. 2, the interval lies well above a value of 0. Note also that the interval excludes a value of 0.48, which is the mean value of the effect size estimates in Table 2.
Next consider the effect size estimate from the Antonio (2001) study. The magnitude of the estimate is 0.30, indicating that the outcome scores for the treatment group students in this study were on average approximately a third of a standard deviation higher than the outcome scores for the control group students. Using the standard error of the estimate (i.e., 0.04) to construct a 95% interval, we see that the lower boundary of the interval is 0.22 and the upper boundary 0.38 (see Table 2, Fig. 2). Just as in the case of the Lopez et al. study, the notion that the true effect size for this study is 0 is not very plausible. But as can be seen the intervals for the Antonio and Lopez et al. studies do not overlap at all, i.e., the interval for the Antonio study lies well below the interval for the Lopez et al. study. This provides evidence that the true effect size for the Antonio study is likely appreciably smaller than the true effect size for the Lopez study.
Visual Displays and Summary Statistics
There are many types of different visual displays available to present and summarize information about magnitudes and distributions of effect sizes (Kosslyn 1994; Tufte 1983). For example, the Tukey’s stemandleaf display is a plot of the data that describes the distribution of results and retains each of the recorded effect sizes (Rosenthal 1995; Tukey 1977). As another example, the schematic plot (i.e., the box plot) records (a) the median effect size, (b) the quartiles, and (c) the minimum and maximum effect sizes (Tukey 1977). The box represents the interquartile range which contains the middle 50% of the effect sizes. The whiskers are lines that extend from the box to the highest and lowest values, excluding outliers. The line across the box indicates the median.
Summary statistics include measures of central tendency as well as measures of variability. Measures of central tendency include the mean (average) and median (the number that divides the distribution into halves). Measures of variability include: the minimum (smallest value), maximum (largest value), Q_{1} (25th percentile), Q_{3} (75th percentile), Q_{3}–Q_{1} (the range for the middle 50%), and standard deviation (measure of dispersion or variation in a distribution). In addition, examining the distance (e.g., in standard deviation units) of the minimum and maximum effect sizes from the mean, Q_{1}, and Q_{3} of the full distribution of effect sizes is a useful start in data analysis for outliers.
Note that the standard error for the effect size estimate from the Katz and Ivey study (i.e., 0.52; see Table 2) is over three times larger than any of the other standard errors in our dataset, resulting in an extremely wide confidence interval. This suggests that the estimate is quite imprecise. This study is unusual in two other respects. It was the only sample in the study published prior to 1990. Secondly, this was a study with an all White sample and an intense intervention (i.e., over two weekends) designed specifically to reduce racism among Blacks. We omitted this study from the first set of HLM analyses presented below, but included it in a second set of analyses to see whether key results were sensitive to the inclusion/exclusion of this case.
Another kind of display known as a forest plot presents the effect sizes graphically with their associated 95% confidence intervals (Light et al. 1994). In a forest plot, each study is represented by a box whose center symbolizes the effect size estimated from that study (randomeffects point estimate) and the lines coming out from either side of the box represents the 95% confidence interval. The contribution of each study to the metaanalysis (i.e., its weight) is represented by the area of the box. The summary treatment effect (average) of all the effect sizes can be found at the very bottom of the forest plot and is shown by the middle of a diamond whose left and right extremes represent the corresponding confidence interval.
Returning to Table 2, we know that connected with each effect size estimate is a certain amount of error that is captured by the standard error of the estimate. One question that arises is: How much of the variability that we see among the effect size estimates is attributable to such error (e.g., lack of precision in each of the estimates), and how much is attributable to actual underlying differences across studies in effect size (i.e., heterogeneity in effect size)? A second question, as noted earlier, is: What factors underlie this heterogeneity? How do differences in various key study characteristics relate to differences in effect size? A key factor that we will focus on in our analyses is the pedagogical approach employed in a study’s program to reduce racial bias. We will also focus on effect size type, i.e., whether or not an effect size is adjusted for possible differences in covariates (e.g., pretest). The following section explains and illustrates how to use HLM analyses to answer these important metaanalytic questions.
HLM Analyses
HLM is an appropriate, effective, and natural technique for quantitative metaanalysis (Raudenbush and Bryk 2002). The data used in quantitative metaanalysis essentially have a nested or hierarchical structure, i.e., subjects (individuals) nested within studies. As noted above, when we consider the variability in effect size estimates for a sample of studies (see, e.g., Table 1, Fig. 2), the variation that we see stems from two sources: the error or lack of precision connected with each study’s effect size estimate, and actual differences across studies in their true effect sizes. HLMs for metaanalysis consist of two interconnected models that enable us to represent and partition these two sources of variation. As will be seen, the information regarding the effect size based on the subjects nested in a study is summarized and represented in a withinstudy (level 1) model; for each study, key elements of the withinstudy model are the estimate of the true effect size for a study and a variance term based on the standard error of the estimate. To represent and investigate the amount of variance across studies in their true effect sizes, and to study factors that systematically underlie such heterogeneity, we pose a betweenstudy (level 2) model.
Raudenbush and Bryk (2002) offer the valuable perspective that more traditional applications of HLM can in a sense be viewed as metaanalyses. For example, consider an application in which we have a sample consisting of students nested within different institutions, and interest centers on the relationship between student involvement in college activities and satisfaction with campus life. Based on Raudenbush and Bryk’s perspective, we can view each institution as providing a study of the relationship between student involvement and satisfaction with campus life. For example, the studentlevel data nested within a given institution would provide information regarding the magnitude of the slope capturing the relationship between student involvement and satisfaction for that institution. A betweeninstitution (level 2) model would enable us to investigate differences in the relationship between student involvement and satisfaction across institutions. Analogous to betweenstudy models, the level 2 model would provide a means of investigating the amount of heterogeneity in involvement/satisfaction slopes across institutions, and institutional characteristics that might underlie this heterogeneity.
This section presents four HLM models: (1) an unconditional model in which we attempt to estimate an overall effect size and the extent to which effect sizes vary around the overall average (N = 29), (2) a conditional model in which differences in effect size are modeled as a function of pedagogical approach (N = 29), (3) a conditional model based in which pedagogical approach and effect size type (i.e., adjusted versus unadjusted) are predictors (N = 29), and (4) a conditional model in which pedagogical approach is the predictor, and the model is fit to a subsample of studies (N = 6).
Model 1
How does HLM use the effect size estimates (g _{ j }) for the set of studies in our sample and the error variances (V _{ j }) connected with these estimates to obtain an estimate of the mean effect size (γ_{0}) and the variance component τ? Returning to Table 2, note that if the error variances for our studies were approximately equal to 0 (i.e., if the g _{ j }’s were extremely accurate estimates of the δ_{ j }’s), that would mean that practically all of the variability that we see in plots of the effect size estimates (i.e., Fig. 1) would be attributable to actual, underlying differences in effect size (i.e., parameter variance). Alternatively, if the error variances were extremely large (e.g., if the error variance for each study was as large as the error variance for the Katz and Ivey study), that would signal that much of the variability that we see in Fig. 1 may very likely be attributable to error variance.
Thus, when we inspect plots such as Fig. 2, we know that the total variance of a sample of effect size estimates (Var (g _{ j })) consists of two components: error variance (V _{j}) and random effects (parameter) variance (τ). What is crucially important is that we have information about the magnitude of error variance. HLM is able to use this information and the g _{ j }’s to compute a maximum likelihood estimate of τ, which we will term \( \hat{\tau } \).
HLM summary of Models 1, 2, 3, and 4
Fixed effect  Model 1^{a}  Model 2^{a}  Model 3^{a}  Model 4^{b}  

Estimate (SE)  tRatio  Estimate (SE)  tRatio  Estimate (SE)  tRatio  Estimate (SE)  tRatio  
Intercept, γ_{0}  0.47 (0.033)  14.21***  0.45 (0.033)  13.69***  0.47 (0.037)  12.82***  0.32 (0.033)  9.80*** 
Pedagogical approach, γ_{1}  0.28 (0.112)  2.52*  0.35 (0.096)  3.70**  0.38 (0.125)  3.05*  
Effect size type, γ_{2}  −0.12 (0.041)  −2.96** 
Random effect  Variance component  χ^{2}  df  Variance component  χ^{2}  df  Variance component  χ^{2}  df  Variance component  χ^{2}  df 

True effect size, δ_{j}  0.027  1171.05***  28  0.025  1155.21***  27  0.024  1145.65***  26  0.000  3.86  4 
The estimate of τ (i.e., the amount of parameter or random effects variance among the effect sizes) is approximately 0.03. Since standard deviations can sometimes be more interpretable measures of spread than variances, we can take the square root of our estimate of τ to obtain a standard deviation capturing the variation in effect sizes (i.e., 0.16). As noted above, our model assumes that true effect sizes are normally distributed around an average effect size with a certain amount of variation. If the normality assumption holds approximately, then based on our results we can use our estimate of the overall mean (0.47) and our measure of variation (0.16) to get a better sense of the extent to which the studies actually differ in their effect sizes.^{2} For example, a study whose effect size is one standard deviation above the mean would be 0.47 + 0.16 = 0.63, while a study whose effect size is two standard deviations above the mean would be 0.47 + 0.32 = 0.79. Similarly, studies whose effect sizes are one and two standard deviations below the mean would be 0.31 and 0.15, respectively. Due to the importance of the estimate of the random effects variance (τ) for assessing whether the studies in a sample may tend to be homogeneous and have a common true effect size, or whether the studies may tend to vary appreciably in their true effect sizes (i.e., whether there is substantial heterogeneity around the average effect size), the null hypothesis (τ = 0) should be tested. HLM’s chisquare test of the hypothesis that τ = 0 yields a pvalue < 0.001, i.e., there is strong evidence against the null hypothesis.^{3}
Model 2
The results for the fixed effects (γ_{0}, γ_{1}) and τ in Model 2 appear in Table 3. Note that the estimates of the fixed effects are obtained via a weighted regression of the effect size estimates for our sample of 29 studies on pgm_type _{ j }, where the weights are \( {\frac{1}{{V_{j} + \hat{\tau }}}} \). Thus those studies with effect sizes that are estimated more precisely receive more weight. As can be seen, the estimate of the expected effect for C studies is 0.45 (SE = 0.03). Moreover, we see that the expected difference between C only studies versus C + CRI studies is approximately 0.28 (SE = 0.11), a difference that is statistically significant. Thus the expected effect size for C + CRI studies is 0.45 + 0.28 = 0.73 (SE = 0.11).^{4} This estimate is very sensible when we consider the magnitude of the effect size estimates for the Lopez et al. (1998), Muthuswamy et al. (2006) and Gurin et al. (2004) studies in Table 2 (i.e., 0.87, 0.86 and 0.48, respectively). Note, finally, that the estimate of τ is approximately 0.025. When we compare this estimate with the estimate from the previous analysis (i.e., 0.027), we see that the inclusion of pgm_type _{ j } in the analysis has resulted in a reduction in parameter variance of approximately 7%.
Model 3
Similar to a weighted multiple regression analysis, the estimates of the fixed effects in Eq. 10 are obtained via a weighted regression of the effect size estimates in our sample on the program type and effect size type, where the weights are \( {\frac{1}{{V_{j} + \hat{\tau }}}} \). Looking at the results for Model 3 in Table 3, we see that the resulting estimate for γ_{2} is −0.12, i.e., holding constant program type, the expected effect size for those studies with adjusted effect sizes tend to be slightly smaller. Moreover, we see that the estimate for γ_{1} (i.e., the expected difference in effect size between C + CRI and C only programs, holding constant effect size type) is slightly larger than the estimate from the previous analysis (i.e., 0.35 (SE = 0.10) vs. 0.28).^{5} However, both this analysis and the previous analysis essentially point to an expected difference in effect size of approximately 0.3 between C + CRI and C only studies.
Model 4
As a final analysis, we focus on the subsample of six studies for which we have adjusted effect sizes, two of which employed C + CRI (i.e., Lopez et al. 1998; Gurin et al. 2004) and four of which employed C only (i.e., Stake and Hoffman 2001 [2 and 3]; Chang 2002; Antonio 2001). Thus the results for these studies have been controlled for potential confounders such as pretest differences. In addition, while the samples of students in the set of 29 studies varied in terms of racial composition (e.g., some samples were composed entirely of white students, some entirely of students of color, and some were a mix of white students and students of color), the composition of samples in the subset of six studies were similar in that they consisted of a mix of white students and students of color.
In this analysis, we employ the same betweenstudy model used in our second analysis, i.e., effect size is modeled as a function of program type (see Eq. 9). The results for Model 4 in Table 3 show that the estimate of the expected effect size for C only studies is approximately 0.32 (SE = 0.03), and that the expected difference in effect size between the C + CRI studies and C only studies is 0.38 (SE = 0.13), which is quite consistent with the results from the previous analysis. In addition, the expected effect size for the C + CRI program type category is: 0.32 + 0.38 = 0.70 (SE = 0.12). Finally note that the resulting estimate of τ is approximately equal to 0, and the test of homogeneity yields a pvalue > 0.50. For this subset of studies, effect size, conditional on program type, appears to be fairly homogeneous. Note that this result is probably driven by the fact that the 4 effect size estimates from the C only studies are highly homogenous, ranging between values of 0.30 and 0.39.
HLM Analyses Summary
Models 2, 3, and 4 all point to the expected effect of programs with a CRI component to be substantially larger than those with contentbased knowledge only. Also, the average or expected effect size for programs that involve both components is, based on all three C + CRI studies, 0.73 with an SE of 0.11 (Model 2); and based on the two C + CRI studies with adjusted effect sizes, the expected effect size is 0.70 with an SE of 0.12 (Model 4). Taking a closer look at these three studies (i.e., Gurin 2004; Lopez et al. 1998; Muthuswamy et al. 2006), they were all conducted at either the University of Michigan or Michigan State University. These studies all examined diversity “programs” which have been implemented on a single campus with the specific purpose of improving the institutional diversity climate. Thus, it appears that the observed differences may be due to a combination of the CRI component and institutional commitment to diversity which makes these interventions/programs successful in reducing racial bias.
Sensitivity Analyses
Every metaanalysis involves a number of decisions to be made that can affect the conclusions and inferences drawn from the findings (Greenhouse and Iyengar 1994). Should outliers be included or excluded? Should a random effects model be used (e.g., an HLM model), or should a fixed effects model be used (i.e., a model that, in contrast to HLM, does not include a betweenstudy variance component (τ))? Does publication bias (e.g., systematic differences in effect sizes for published versus unpublished studies) pose a threat to the findings? Because of the many possibilities, it is useful to carry out a series of sensitivity analyses to assess whether the assumptions or decisions in fact have a major effect on the results of the review. In other words, are the findings robust to the methods used to obtain them (Cochrane Collaboration 2002)? Thus, sensitivity analyses involve comparing two or more metaanalytic findings calculated based on different assumptions or approaches. Three sensitivity analyses were conducted: (1) inclusion versus exclusion of outliers, (2) random effects versus fixed effects models, and (3) possible publication bias (i.e., published versus unpublished studies).
Inclusion Versus Exclusion of Outliers
Statistical procedures used to determine the significance of an effect size summary statistic are typically based on certain assumptions, for example, that there is a normal distribution of effect sizes surrounding the true mean effect size. If these assumptions are violated, they can influence the validity of the conclusions drawn from the metaanalytic findings. One of the main purposes of the descriptive results was to examine the distribution of effect sizes by presenting visual displays, measures of central tendency, and measures of variability. From these various plots, there was one obvious outlier which was the only study in the metaanalysis that was published prior to 1990 (i.e., Katz and Ivey 1977). While this study had one of the smallest samples (n = 24), it had the largest effect size (d = 2.17) which was more than twice as large in magnitude as the next largest effect size. Upon closer inspection, however, this finding was not entirely surprising. First, the study was conducted in the 1970s, long before diversity research really took off. Second, the study participants were all White. Thus, it was apparent that this study was an outlier for a variety of reasons and was eliminated from the inferential analyses.
Comparison of Models 1 and 3 (without and with outlier)
Model 1: Without outlier (n = 29)  Model 1: With outlier (n = 30)  

Fixed effect  Coefficient  S.E.  tRatio  95% CI  Fixed effect  Coefficient  S.E.  tRatio  95% CI 
Grand mean, γ_{0}  0.47  0.033  14.21***  (0.41, 0.54)  Grand mean, γ_{0}  0.48  0.034  14.17***  (0.41, 0.54) 
Random effect  Variance component  df  χ^{2}  PVR  Random effect  Variance component  df  χ^{2}  PVR 

True effect size, δ_{j}  0.027  28  1171.05***  (0.15, 0.79)  True effect size, δ_{j}  0.028  29  1192.46***  (0.15, 0.80) 
Model 3: Without outlier (n = 29)  Model 3: With outlier (n = 30)  

Fixed effect  Coefficient  S.E.  tRatio  95% CI  Fixed effect  Coefficient  S.E.  tRatio  95% CI 
Intercept, γ_{0}  0.47  0.037  12.82***  (0.40, 0.54)  Intercept, γ_{0}  0.48  0.037  12.82***  (0.41, 0.55) 
Pedagogical approach, γ_{1}  0.35  0.096  3.70**  (0.17, 0.54)  Pedagogical approach, γ_{1}  0.35  0.095  3.68**  (0.16, 0.54) 
Effect size type, γ_{2}  −0.12  0.041  −2.96**  (−0.20, −0.04)  Effect size type, γ_{2}  −0.13  0.041  −3.10**  (−0.21, −0.05) 
Random effect  Variance component  df  χ^{2}  PVR  Random effect  Variance component  df  χ^{2}  PVR 

True effect size, δ_{j}  0.024  26  1145.65***  (0.17, 0.78)  True effect size, δ_{j}  0.025  27  1166.90***  (0.17, 0.79) 
On a related note, the Vogelgesang (2000) study may also be seen as an outlier of sorts because this study contributed nine effect sizes to the analyses (ranging from 0.25 to 0.43). Thus, it is also a good idea to examine what sort of impact the information from her studies is having on the results. In Model 4 where we focus only on those studies that involved matching or covariate adjustment, the effect sizes from her studies are set aside, but we still see a substantial difference in pedagogical approach between the C + CRI and C only studies. Thus, Model 4 may also be viewed as a sensitivity analysis as well.
Comparison of Models 1 and 3 (with all 9 Vogelgesang effect sizes and with an averaged Vogelgesang effect size)
Model 1: With all 9 Vogelgesang effect sizes (n = 29)  Model 1: With all 9 Vogelgesang effect sizes (n = 21)  

Fixed effect  Coefficient  S.E.  tRatio  95% CI  Fixed effect  Coefficient  S.E.  tRatio  95% CI 
Grand mean, γ_{0}  0.47  0.033  14.21***  (0.41, 0.54)  Grand mean, γ_{0}  0.52  0.042  12.56***  (0.44, 0.60) 
Random effect  Variance component  df  χ^{2}  PVR  Random effect  Variance component  df  χ^{2}  PVR 

True effect size, δ_{j}  0.027  28  1171.05***  (0.15, 0.79)  True effect size, δ_{j}  0.030  20  1288.80***  (0.19, 0.86) 
Model 3: With all 9 Vogelgesang effect sizes (n = 29)  Model 3: With an averaged Vogelgesang effect size (n = 21)  

Fixed effect  Coefficient  S.E.  tRatio  95% CI  Fixed effect  Coefficient  S.E.  tRatio  95% CI 
Intercept, γ_{0}  0.47  0.037  12.82***  (0.40, 0.54)  Intercept, γ_{0}  0.54  0.036  14.91***  (0.47, 0.61) 
Pedagogical approach, γ_{1}  0.35  0.096  3.70**  (0.17, 0.54)  Pedagogical approach, γ_{1}  0.33  0.096  3.43**  (0.14, 0.52) 
Effect size type, γ_{2}  −0.12  0.041  −2.96**  (−0.20, −0.04)  Effect size type, γ_{2}  −0.19  0.049  −3.77**  (−0.28, −0.09) 
Random effect  Variance component  df  χ^{2}  PVR  Random effect  Variance component  df  χ^{2}  PVR 

True effect size, δ_{j}  0.024  26  1145.65***  (0.17, 0.78)  True effect size, δ_{j}  0.024  18  1374.30***  (0.23, 0.84) 
Random Effects Versus Fixed Effects
Comparison of Models 1 and 3 (random effects versus fixed effects)
Model 1: Random effects model  Model 1: Fixed effects model  

Fixed effect  Coefficient  S.E.  tRatio  95% CI  Fixed effect  Coefficient  S.E.  tRatio  95% CI 
Grand mean, γ_{0}  0.47  0.033  14.21***  (0.41, 0.54)  Grand mean, γ_{0}  0.45  0.005  95.19***  (0.45, 0.46) 
Random effect  Variance component  df  χ^{2}  PVR  Random effect  Variance component  df  χ^{2}  PVR 

True effect size, δ_{j}  0.027  28  1171.05***  (0.15, 0.79)  True effect size, δ_{j}  –  –  –  – 
Model 1: Random effects model  Model 1: Fixed effects model  

Fixed effect  Coefficient  S.E.  tRatio  95% CI  Fixed effect  Coefficient  S.E.  tRatio  95% CI 
Intercept, γ_{0}  0.47  0.037  12.82***  (0.40, 0.54)  Intercept, γ_{0}  0.46  0.005  94.48***  (0.45, 0.47) 
Pedagogical approach, γ_{1}  0.35  0.096  3.70**  (0.17, 0.54)  Pedagogical approach, γ_{1}  0.39  0.108  3.62***  (0.18, 0.60) 
Effect size type, γ_{2}  −0.12  0.041  −2.96**  (−0.20, −0.04)  Effect size type, γ_{2}  −0.14  0.029  − 4.77***  (−0.20, −0.08) 
Random effect  Variance component  df  χ^{2}  PVR  Random effect  Variance component  df  χ^{2}  PVR 

True effect size, δ_{j}  0.024  26  1145.65***  (0.17, 0.78)  True effect size, δ_{j}  –  –  –  – 
File Drawer Problem: Publication Bias Problem
The final sensitivity analysis concerns the “file drawer problem”—a problem that threatens every metaanalytic review (Lipsey and Wilson 2001; Rosenthal 1979, 1991). The file drawer problem is a criticism of metaanalysis in that the available studies to the researcher(s) may not be representative of all the studies ever conducted on a particular topic (Becker 1994; Greenhouse and Iyengar 1994; Rosenthal 1979, 1991). The most likely explanation for this is that researchers may have unpublished manuscripts tucked away in their “file drawers” because their results were not statistically significant and were therefore never submitted for publication. Thus, publication bias arises when studies reporting statistically significant findings are published whereas studies reporting less significant or nonsignificant results are not. As a result, many metaanalyses may systematically overestimate effect sizes because they rely mainly on published sources. Of greatest concern is whether the file drawer problem is large enough in a given metaanalysis to influence the conclusions drawn (Lipsey and Wilson 2001). Consequently, it is suggested that mean comparisons, sample size and effect size comparisons, as well as calculation of a failsafe N were employed in order to examine this possibility (Light and Pillemer 1984; Lipsey and Wilson 2001).
Rosenthal’s (1979)/Orwin’s (1983) failsafe N. A final approach is to test publication bias empirically. Rosenthal (1979) developed a failsafe N that can be calculated in order to estimate possible publication bias. This failsafe N estimate represents the number of additional studies with nonsignificant results that would have to be added to the sample in order to change the combined p from significant (at the 0.05 or 0.01 level of confidence) to nonsignificant (Rosenthal 1979). This failsafe N was developed for use with Rosenthal’s method of cumulating zvalues across studies. This failsafe N formula estimates the number of additional studies needed to lower the cumulative z below the desired significance level, for example, a z equal to or less than 1.645 (p ≥ 0.05). However, Rosenthal’s (1979) failsafe N is limited to probability levels only.
The failsafe N’s utility is similar to that of the confidence interval (Carson et al. 1990). The failsafe N provides additional information on the degree of confidence that can be placed in a particular metaanalytic result, in this case, the overall mean effect of curricular and cocurricular diversity activities on racial bias in college students. An important difference, however, is that the failsafe N assumes that the unobserved studies (i.e., those tucked away in a file drawer somewhere) are assumed to have a null result. For example, if the failsafe N is relatively small in comparison to the number of studies in the metaanalysis, caution should be used when interpreting metaanalytic results. On the other hand, if the failsafe N is relatively large in comparison to the number of included studies, then more confidence is assured regarding the stability of the results. Rosenthal (1979) suggests as a general rule a reasonable tolerance level of 5k + 10. If the failsafe N exceeds the 5k + 10 benchmark, then a filedrawer problem is unlikely.
Study Limitations and Implications
One of the important outcomes of every metaanalysis is a set of implications and recommendations for the conduct and reporting of future work in a given domain. As such, this section presents some guidelines and recommendations for reporting of results and designing of studies to enable and enhance future metaanalytic reviews.
Limitations Connected with the Reporting of Results
A relatively small number of effect sizes (29) were included in the study, which was due in part to limitations in the reporting of results. While many studies made an effort to statistically control for prior differences in various covariates (e.g. pretest), there was a lack of adequate information for computing adjusted effect sizes and/or their error variances. In our analyses, we saw that the unadjusted effect sizes tended to be only slightly larger, on average, than the adjusted effect sizes. Furthermore, we saw that the programs with both contentrelated knowledge and crossracial interaction appear to have larger effect sizes than the contentrelated knowledge only studies when we control for effect size type, and when we focus only on the subset of studies for which we have adjusted effect sizes. Yet it is important that studies try to report the information that would be necessary for computing adjusted effect sizes and their error variances.
There is also a lack of detailed information regarding the diversityrelated interventions themselves. Many of the characteristics of the curricular and cocurricular diversity activities themselves were chosen to be coded such as intensity of crossracial interaction (random, builtin), duration of intervention (in weeks), and intensity of intervention (in hours/week). These characteristics were chosen in the hopes of exploring why some diversityrelated activities are more effective than others. However, many of the singleinstitution studies did not report this information in sufficient detail and many of the multiinstitution studies relied on survey data which only provided information about whether or not they participated in a certain type of diversity activity. Thus, unfortunately not many characteristics of the curricular and cocurricular diversity activities themselves could be explored as possible moderators in this metaanalysis.
In both of their books on How College Affects Students, Pascarella and Terenzini (1991, 2005) attempted to conduct “mini” metaanalyses whenever possible. However, they quickly realized that a substantial percentage of studies simply did not report sufficient information in order to be able to compute effect sizes. Smart (2005) describes seven attributes that distinguish exemplary manuscripts from other manuscripts that utilize quantitative research methods. One of Smart’s (2005) recommendations for exemplary manuscripts is to report evidence that permits others to reproduce the results, such as means, standard deviations, and correlations among all the variables included in the analyses at the very minimum. These simple but informative descriptive statistics allow calculation of effect sizes for future metaanalytic reviews.
Limitations Connected With the Design of Studies
In the case of this illustrative metaanalysis of the effects of diversity programs, programs that incorporated both contentrelated knowledge and crossracial interaction emerge as being very promising. Given the small number of such studies, additional research on the effects of these programs is needed. In connection with this, it would be valuable for researchers to conduct studies that directly compare contentbased knowledge programs with programs that include the additional crossracial interaction component.
Many of the studies could also benefit from increased research rigor. Due to the nature of this research, it is unsurprising that the current literature is overwhelmingly correlational. As such, there is concern over threats to internal validity (Shadish et al. 2002). In other words, how do we know that the curricular or cocurricular diversity activity is responsible for the change(s) seen in students’ racial bias? One of the major shortcomings of these studies is that there is clearly selfselection of the students into these curricular and cocurricular diversity activities. While randomly assigning students to content only or content plus crossracial interaction programs might not be feasible, efforts should be made to construct these two groups of students that are as comparable as possible at the outset (e.g., Shadish et al. 2002). For example, greater care should be taken to choose comparison groups that are similar in important ways to the groups of program participants and to utilize a rich set of covariates to adjust for initial differences.
Conclusion
In this article, we have demonstrated how this powerful analytical technique can be used to quantitatively synthesize the research findings on a specific research topic. HLM plays a key role as a powerful and natural methodology for a broad class of metaanalytic phenomena. Complementing qualitative literature reviews, metaanalysis—and particularly metaanalysis utilizing HLM—provides a quantitative “big picture” view of the available knowledge base as well as explanations for the similarities and differences among the various studies (Rosenthal and DiMatteo 2001). While originally developed to assess the overall average effectiveness of a program or intervention, what is exciting are the moderator analyses within metaanalysis that can explain the factors that diminish or amplify the effect sizes. No matter what the topic—from the effects of service learning to the effects of university outreach programs—as the research accumulates oftentimes it becomes increasingly difficult to make sense of the literature. It is at this stage when an HLM metaanalysis would be valuable. By providing a brief overview and an illustrative example of metaanalysis utilizing HLM in higher education, we hope to encourage more widespread use of this methodological approach in the field.
Footnotes
 1.
For the remainder of the article, the subscript “j” will be used to index or denote individual studies in the sample.
 2.
See Raudenbush and Bryk (2002, pp. 274–275) for a method for assessing the reasonableness of this assumption, and for a discussion of alternative estimation approaches when the normality assumption may appear untenable.
 3.
This test is conceptually similar to the homogeneity test based on the Q statistic in other metaanalytic procedures.
 4.
The standard error for the expected effect size for C + CRI studies, can be found by squaring the standard error of the estimate for γ _{0} (SE = 0.03), squaring the standard error of the estimate of γ_{1} (SE = 0.11), and proceeding as follows:
\( {\text{SE of the expected C}} + {\text{CRI effect size}} = \sqrt {(0.03)^{2} + (0.11)^{2} } \)
 5.
The larger estimate of 0.35 can be viewed as a way of statistically compensating for the fact that adjusted effect sizes tend to be smaller than unadjusted ones. While 2 of the 3 C + CRI studies have adjusted effect sizes, only 4 of the 26 C studies have adjusted effect sizes.
 6.
Technically the logic of funnel plots is rooted in the notion that the studies in a sample are homogeneous, i.e., that they vary only in terms of error variance and share a common, overall effect size. It is often the case, however, that effect sizes are heterogeneous. In such situations, there is the possibility that a funnel plot may have what appears to be a “bite” taken out of the lower left portion of the funnel, but the pattern may not be due to publication bias. For example, if programs are implemented more carefully or more intensively in studies with small sample sizes, such studies may tend to have larger effect sizes compared with largescale studies; in that case, the distribution of effect sizes for small studies would be shifted toward the right of the plot, leaving what appears to be a “bite” in the lower left portion of the plot (Shadish, personal communication, February 2010; Sutton 2009). Thus, more generally it is important to check if study sample size is systematically related to study characteristics that might moderate the magnitude of effects.
Notes
Acknowledgments
The authors wish to thank Ernest T. Pascarella and the two anonymous reviewers for their valuable feedback and thoughtful comments in the writing of this manuscript.
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