1. Conducting a Meta-Analysis Jose Ramos Southern Methodist University Paper presented at the annual meeting of the Southwest Educational Research Association, San Antonio, TX, February 2-4, 2011.

2. Outline Development & Theory Other Methods: Simpson’s Paradox Methodological Considerations Calculating Effect Sizes Combining Effect Sizes Across Studies Analyzing Variance in Effect Sizes Activity

3. Development & Theory Development of Meta-Analysis  Research synthesists’ methods employed idiosyncratic rules to reach conclusions  Individuality did not allow for a streamlined method of properly synthesizing data (i.e., no standardization)  Narrative approach to synthesizing literature was subjective What is a meta-analysis?  A standardized secondary analysis of primary data results from different studies that share same hypothesis  A quantitative aggregation of findings during a research synthesis  Calculating a standardized effect size for multiple studies

4. Development & Theory What is an effect size?  It is “how much?”  Called the magnitude of difference or relationship strength (i.e., dependent variable)  The degree to which the phenomenon is present in the population  * Is some specific NON-ZERO value in the population  “The greater this value, the greater the degree to which the phenomenon under study is manifested” (Cooper, 2010, p. 162).  How does this differ from just saying that the results were “statistically significant”?

5. Development & Theory Why do we need meta-analyses?  Literature expansion in social science research  Allows researchers the ability to statistically combine countless studies to increase power  Allows us to measure “how much” of a relationship exists, rather than just whether a relationship exists  Allows us to account for the variation in results between similar studies based on procedural characteristics of individual studies

6. Voting Method Voting method was commonly employed for aggregation of studies before the conception of meta-analysis Procedure:  Studies with a dependent variable and a specific independent variable are examined  Studies are dichotomized as either statistically significant or not statistically significant  Classification with higher tally is considered to be the “true” relationship between variables

7. Voting Method Flaws:  Bias in favor of large-sample studies Why is this a problem?  No weighting of sample size  Tells us nothing about strength of relationship  Does not control for variation between studies

8. Voting Method Illustration:  Researcher A is conducting a study on the effects of RtI on a group of 1 st graders’ fluency rate.  In A’s study, which has a sample size of n=180, 110 children are given RtI and 70 children are given traditional instruction. After 12 weeks of instruction, children are dichotomized as either “pass” or “fail” on a reading measure.  The improvement rate for the RtI group is.45 vs..43 for the control group.

9. Voting Method Illustration:  Researcher B conducts the same study at a different site  In B’s study, which has a sample size of n=230, 90 children receive RtI and 140 receive traditional instruction  Again the improvement rate for the RtI group is.67 vs..64 for the control group.  That’s 2-0 for the experimental group!

10. Aggregation of Raw Data Suppose another researcher aggregates the data from the same studies by summing the raw data instead of employing the voting method Illustration:  Add the number of subjects in both studies that received treatment and control: n=200 received RtI and n=210 received traditional instruction  When dichotomized into “pass” or “fail”, the improvement rate for the treatment group is now ? vs. ? for the control group!  This is known as Simpson’s Paradox

11. Simpson’s Paradox Occurs when different methods of aggregation are employed on the same studies Leads to opposing conclusions Has nothing to do with statistical significance Can occur in a meta-analysis So, why does this happen?  Unbalanced experimental designs

12. Methodological Considerations Was an appropriate method used to combine and compare results across studies? Were the variables and relationship of interest defined? Are the studies included address an identical conceptual hypothesis Was an extensive search of the literature done? Where could we look?

13. Methodological Considerations Determine the statistic of interest to calculate individual study effect sizes:  Is your hypothesis assessing the relationship between a dichotomous and continuous variable? Two continuous variables? Two dichotomous variables?  What do the preponderance of your studies report as an effect size, if any?  Based on this information you will choose one standardized effect size: r, d, or odds-ratio in your meta-analysis

14. Calculating Effect Sizes d-index:  Appropriate to use when the difference between two means is being compared; a dichotomous and continuous variable  Typically employed in association with t- or F- tests, based on a comparison of two conditions  Expresses the distance between the two group means in relation to their common SD

15. Calculating Effect Sizes d-index formula:, where:

16. Calculating Effect Sizes So, if you were to calculate the standardized mean difference in the fluency rate of the following two groups in an RtI study, what would you get as the effect size?  Group 1 (experimental): M 1 = 80, SD 1 =10, n 1 = 250  Group 2 (control): M 2 = 65, SD 2 = 20, n 2 =230  Effect size = ? What if you had three groups?

17. Calculating Effect Sizes What if the means and SDs aren’t reported and you only have a t-value?  Formula for d-index when only t-value is reported: where: df error = (n 1 +n 2 -2) What if you have the F-value for two means?  Formula for d-index when the F-value of two means is reported:

18. Calculating Effect Sizes What is the F-value is based on 3 or more means and I want to get the d-index?  You must first consider the dispersion of the means to identify the pattern of variability: minimum, intermediate, or maximum.  Less variability between means will give you a lower F-value.  This will dictate which of three formulas you will use.

19. Calculating Effect Sizes Formula for minimum variability: Formula for intermediate variability:  Formula for maximum variability:  d = 2f (even number of means)  (odd number of means)

20. Calculating Effect Sizes r-index  The correlation coefficient tells you about the strength of the relationship between two variables  Most appropriate metric for expressing an effect size when interested in the relationship strength of two continuous variables  Most common in correlational studies  Usually reported when appropriate  EX: relationship between years of schooling and yearly salary

21. Calculating Effect Sizes What if you only have a t-value?  Formula for r-index when only t-value is reported:,where: df error = (n 1 +n 2 -2) What if you only have an F-value?  In an ANOVA, Pearson’s r is similar to η (eta)  In an ANOVA, Pearson’s r 2 is similar to η 2 (eta-squared)  Recall that η 2 (eta) tells us the variance in the dependent variable that is explained by the independent variables

22. Calculating effect size Formula to convert F-value to η (i.e., r): Formula to convert F-value to η 2 (i.e., r 2 ):

23. Calculating Effect Sizes Think back to the previous RtI study on slide 16. The effect size was d = 0.96. The difference between the control/experimental group. Suppose you want to convert this d-index into an r-index:, where  What do you get?  r = ?  What could this correlation represent? Or vice-versa:

24. Calculating Effect Sizes Odds-Ratio (OR)  Applicable when both variables are dichotomous  The relationship between two sets of odds  EX: Suppose a study measures the effects of RtI on whether students in two groups (e.g., experimental/control ) “pass” or “fail” a math test.  RtIControl Pass75 (a)40 (b) Fail5 (c)25 (d)

25. Calculating Effect Sizes Of n=80 in RtI, the ratio of passing is 15 to 1. Of n =65 in control, the ratio of passing is 1.6 to 1. Calculate the odds ratio: OR = ad/bc = ?

26. Combining Effect Sizes Once individual study effect sizes have been calculated, the next step involves combining them to provide an average effect size. You must weight the individual effect sizes.  What do you base this weight on?

27. Combining Effect Sizes Suppose you have 7 d-indexes and group ns that compares the effect of homework vs. no homework on a measure of academic achievement:

28. Combining Effect Sizes Step One: Weighting Formula: EX: Study 1 w i = 2(259 + 265) 259* 265/2(259 + 265) 2 + 259* 265*.02 2 = 130.98

29. Combining Effect Sizes Calculations:

30. Combining Effect Sizes Step Two: Multiply each weighted effect size and original d-index Formula: d i w i EX: What is the answer for Study 1?

31. Combining Effect Sizes Calculations:

32. Combining Effect Sizes Step Three: Divide the sum of these products by the sum of the weights. Formula: EX: d. = 62.56/544.27 = +.115 (average ES)

33. Combining Effect Sizes Step Four: Computing Confidence Intervals Formula: EX:  Thus, we expect 95% of estimates of this effect to fall between.031 and.199. Do we reject the null?

34. Combining Effect Sizes Suppose that you have 6 r-indexes and ns that show the relationship between the amount students spend on homework and their score on an achievement test. Step One: Transform the r-indexes into a z-scores because as r gets larger the distribution gets more skewed. Formula:

35. Combining Effect Sizes Step Two: Weighting Formula: n i - 3 EX: Study 1 3,505-3 = 3,502

36. Combining Effect Sizes Calculations:

37. Combining Effect Sizes Step Three: multiply the weight and the effect size (i.e., z-score) Formula: (n i – 3) z i EX: Study 1 (3,502).06 = 210.12

38. Combining Effect Sizes Calculations:

39. Combining Effect Sizes Step Four: Divide the sum of these products by the sum of the weights. Formula: EX: z. = 5462.97/26,372 = +.207 (average ES)

40. Combining Effect Sizes Step Five: Computing Confidence Intervals Formula: EX: CI z95% =.207 ± 1.96/ √26,372 =.207 ±.012  Thus, we expect 95% of estimates of this effect to fall between.195 and.219. Do we reject the null?

41. Analyzing Variance in ES Fixed-effect model vs. random-effect model when accounting for error (i.e., variation) between ES Random-effect model  Two sources of variability in outcomes of similar studies  sampling of participants will always vary from true population  Differences in how studies are conducted  Remember, you sample studies for your meta-analysis from a population of all possible studies  EX: studies on academic achievement based on the level of homework a student does are conducted with students at different grade levels, with different types of tests, in classrooms with different subject matter

42. Analyzing Variance in ES Homogeneity Analyses  Compares the observed variance of the effect sizes to that expected from sampling error  calculates the probability that the variance exhibited by the effect sizes would be observed if only sampling error was making them different  So the question is “Is the observed variance in effect sizes statistically significantly different from that expected by sampling error alone?” (Cooper, 2010, p. 185)  If it is, you search for moderator variables

43. Activity Suppose your question is: “What is the effect of RtI on the vocabulary development of ELLS?” and you find the following studies: Study 1: Data on the effects of RtI on the vocabulary development of 3 rd grade ELLs:  Study 1  n 1 = 120 (RtI); n 2 = 115 (control)  M 1 = 65; M 2 = 50  SD 1 = 15; SD 2 = 30  ES = ?

44. Activity Study 2: The relationship between the length of time 2 nd grade ELLs students spend in RtI and the number of vocabulary words they learn:  Study 2  r =.30  n 1 = 100; n 2 = 110  ES = ? Study 3: A t-value on the difference between an RtI group (experimental) vs. a control group in 4 th grade bilingual classroom:  Study 3  t-value: 3.2  n 1 = 80; n 2 = 85  ES = ?

45. Activity Study 4: The standardized mean difference of two 4 th grade ELL groups on vocabulary development, one in a RtI treatment and one in a control group:  Study 4  d =.55  n 1 = 65; n 2 = 60  ES = ? Study 5: Data on the effects of RtI on the vocabulary development of 7 th grader ELLs:  Study 5  n 1 = 150 (RtI); n 2 = 140 (control)  M 1 = 25; M 2 = 18  SD 1 = 5; SD 2 = 10  ES = ?

46. Activity Step 1: Decide which standardized effect size is most appropriate to calculate an average effect size. Step 2: Standardize effect sizes Step 3: Calculate average effect size Step 4: Calculate 95% Confidence intervals Step 5: Do we reject the null?

47. References Blyth, C. R. (1972). On Simpson’s paradox and the sure-thing principle. Journal of the American Statistical Association, 67, 364-366. Cohen, J. (1988). Statistical power analysis for the behavioral sciences. (2 nd Ed.). New Jersey: LEA. Cooper, H. (2010). Research synthesis and meta-analysis: A step-by- step approach. (4 th Ed.). New York: SAGE. Cooper, H., Hedges, L. V., & Valentine, J.C. (1994). The handbook of research synthesis and meta-analysis. (2 nd Ed.). New York: SAGE. Glass, G. V (1976). Primary, secondary, and meta-analysis of research. Educational Researcher, 5, 3-8. Glass, G.V (1977). Integrating findings: The meta-analysis of research. Review of Research in Education, 5, 351-379.

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