Univariate and Multivariate Power Analysis

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Other Programs:  | Semipartial power analysis | p-values from test statistics | effect size from test statistics |

Program Description

The power analysis performed in this application is based on a generalized method suitable for virtually any parametric analysis that relies on either the t distribution or the F distribution as a test  statistic.  A-priori power analysis requires that one estimate several values that would be anticipated in the designed study under consideration.  The method requires an anticipated value of Eta-square or R-square (0²,  r² or R²), the degrees of freedom for hypothesis (df_h), the degrees of freedom for error (df_e), and the Type I error rate (alpha) at which statistical tests will be performed.  The program returns the power of the proposed analysis given the values submitted.  The methods employed here are closely allied to those described in Cohen (1988), O'Brien & Muller (1993),  Muller & Peterson (1984) and Murphy & Moyers (1998).  Additional discussion of power analysis and power analytic computational methods can be found in Odeh & Fox (1991), Lipsey (1990), and Kraemer & Thiemann (1987).  The algorithms employed to calculate power values are based on algorithms suggested by Cohen (1988), Abramowitz & Stegun (1977), Cooke, Craven, & Clarke (1982), and SPSS (1991).

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References

Abramowitz, G. G., & Stegun, A. A.  (1972).  Handbook of mathematical functions.  NY:  Dover.

Cohen, J.  (1988).  Statistical power analysis for the behavioral sciences.  Hillsdale, NJ:  Lawrence Erlbaum Associates.

Cohen, J., Cohen, P., West, S.G., & Aiken, L.S.  (2003).  Applied multiple correlation/regression analysis for the behavioral sciences.  Mahwah, NJ:  Lawrence Erlbaum Associates.  (New addition of Cohen & Cohen (1983) of the same title).

Cooke, D. Craven, A. H., & Clarke, G. M.  (1982).  Basic statistical computing.  London:  Edward Arnold Publishers, Ltd. 

Fleiss, J.  (1994).  Measures of effect size for categorical data.  In H. Cooper and L. V. Hedges (Eds.).  The handbook of research synthesis.  NY:  Russel Sage Foundation.

Haase, R. F.  (1991).  Computational formulas for multivariate strength of association from approximate F and X² tests.  Multivariate Behavioral Research, 26(2), 227-245.

Haase, R. F.  (1993).  An SPSS matrix program for computing univariate and multivariate power analysis.  Applied Psychological Measurement, 17, 295.

Haase, R. F.  (2000).  MAT_HYP:  An SPSS matrix language program for testing complex univariate and multivariate general linear hypotheses from matrix data input.  Applied Psychological Measurement, 24, 256.

Haase, R. F., & Ellis, M. V.  (1987).  Multivariate analysis of variance.  Journal of Counseling Psychology, 34(4), 404-413.

Hull, C. H, & Nie, N. H.  (Eds)  (1981).  SPSS update 7-9.  Chicago:  SPSS, Inc.

Kraemer, H. C., & Thiemann, S.  (1987).  How many subjects?  Newbury Park, CA:  Sage Publications.

Lipsey, M. W.  (1990).  Design sensitivity.  Newbury Park, CA:  Sage Publications.

Muller, K. E., & Peterson, B. L.  (1984).  Practical methods for computing power in testing the multivariate general linear hypothesis.  Computational Statistics and Data Analysis, 2, 143-158.

Murphy, K. R., & Myors, B.  (1998).  Statistical power analysis.  Mahweh, NJ: Lawrence Erlbaum Associates.

O'Brien, R. G., & Muller, K. E.   (1993).  Unified power analysis for t-tests through multivariate hypotheses.  In L. K. Edwards (Ed).  Applied analysis of variance in behavioral science.  NY:  Marcel Dekker, Inc.

Odeh, R. E., & Fox, M.  (1991).  Sample size choice.  NY:  Marcel Dekker, Inc.

Olson, C. L.  (1976).  On choosing a test statistic in multivariate analysis of variance.  Psychological Bulletin, 83, 579-586.

Rosenthal, R.  (1994).  Parametric measures of effect size.  In H. Cooper and L. V. Hedges (Eds).  The handbook of research synthesis.  NY:  Russel Sage Foundation.

SPSS, Inc.  (1991).  Statistical algorithms.  Chicago: SPSS, Inc.

Tabachnick, B. G., & Fidell, L. S.  (2001).  Using multivariate statistics.  Boston:  Allyn & Bacon. 

Venter, A., & Maxwell, S. E.  (2000).  Issues in the use and application of multiple regression analysis.  In  H. E. A. Tinsley & S. D. Brown (Eds).  Handbook of applied multivariate statistics and mathematical modeling.  NY:  Academic Press.

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The Basic Ideas of Power Analysis

     The concept of power analysis resides in the ability to estimate the probability of a statistically significant result given a number of anticipated conditions.  To perform a power analysis a-priori, one must estimate three things:  the effect size, the degrees of freedom for the hypothesis, and the degrees of freedom for error.  The degrees of freedom for hypothesis and error are closely associated with the sample size (N) in an analysis, and hence are proxies for deciding on the sample size necessary to have a known probability of rejecting a false null hypothesis (the definition of power).  Every test statistic that is routinely used in data analysis consists of two constituent parts:  (1) an effect size (variously defined; see Cohen, 1988) which documents the size of the effect to be anticipated in the study, and  (2) the size of the study which involves the sample size or some function of the sample size (such as df_e).  Statisticians distinguish between two kinds of probability distributions that underlie test statistics such as t or f or X ²:  the central distribution (which is the probability distribution of the test statistic under the truth of the null hypothesis), and the noncentral distribution (the probability distribution of the test statistic under the truth of the alternative hypothesis).  For any given statistic (such as F, t, X ²), if the null hypothesis is true, then the sampling distribution under this condition centers around a central value of zero (hence the null distribution).  However, if the null hypothesis is false, the sample you draw for your study is not sampled form the null distribution, but rather is sampled form the alternative distribution with a central value that is not zero.  The extent to which these two distributions overlap is the fundamental idea of statistical power.  The central and noncentral distributions overlap less and less as (1) the effect size becomes larger and larger; and (2) as sample sizes grows larger and larger.  Other factors that affect power, such as design features that minimize error variance, the reliability of measures, and so forth,  are presumed to be held constant in this instance, but can also affect the power of the analysis.  Hence as the null and alternative (i.e., central and noncentral distributions) become further and further apart the probability of finding a sample from the alternative distribution which will be declared as "significantly different" from the null distribution rises accordingly  The magnitude of the obtained test statistic is directly related to this non-overlap of the central and noncentral distributions--the power of the analysis (the probability of rejecting a false H₀) increases as the two distributions overlap less and less.  A single value, called a noncentrality parameter, can be used to quantify the extent to which the null and alternative distributions overlap.  The noncentrality parameter for the t-test is simply the value of t itself.  Thus the connection between the test statistic and power is this:  as t gets larger and larger (due to increases in either effect size, sample size, or both) the noncentrality parameter also grows accordingly, the null and alternative distributions overlap less and less, and statistical power goes up commensurately.  For a two-tail, test of the equality of the means of two groups we have,

where Sc is the pooled standard deviation of the two groups.  In order to more clearly illustrate the connection between effect size, sample size, the value of t, and statistical power, the expression in Equation 1 can be rewritten into a two-part sequence where t is a function of (1) an effect size (standardized mean differences), and (2) the size of the study (sample size n₁ and n₂),

     Since the t-test is also the non-centrality parameter, the larger the value of t, the less the central and noncentral distributions overlap, and the higher the power of the analysis (large t's are more "significant" and lead to smaller p values than do small t's for the same df).

     The fundamental task  in performing an a-priori power analysis is to make a good guess as to the quantities in the left and right brackets of Equation 2.  In essence, you must guess as to how your data will turn out after the study is done.  Once this information is at hand, one can investigate the power of the analysis as a function of varying the two elements of the noncentrality parameter--effect size and sample size.

     The noncentrality parameter for the F-test is quite similar in nature to the t-test.  As a very general test statistic, the F-ratio can also defined as a two-part expression:  (1) the ratio of the hypothesis sum of squares (SS_h) to the error sum of squares (SS_e) which defines one version of the effect size, and (2) the ratio of the error degrees of freedom (df_e) to the hypothesis degrees of freedom (df_h), which is the part of the expression that carries information about the size of the study.  Note that the degrees of freedom for error is a proxy for sample size in that df_e = N - q_f - 1, where N is the sample size, and q_f is the number of predictor vectors in the full model (see Cohen & Cohen (1983) or Cohen, Cohen, West, & Aiken (2003) for an explanation of full and restricted models).  This form of the F-test can be written as, 

     To perform an a-priori power analysis where the F-test is the test statistic, one must estimate both expressions on the right hand side of the equal sign in Equation 3.  The effect size is documented by the SS_h/SS_e ratio, and the size of the study is documented by the df_e/df_h ratio.  Most investigators do not think in sums of squares terms when considering the magnitude of effects in their research.  There is no scaled metric for SS; they are bounded by zero on the low end, but can take on maximum values that are scale-dependent.  Hence the notion of how big a SS_h must be to constitute a "small", "medium", or "large" effect is scale dependent and varies from study to study.  Most investigators, however, are quite used to thinking in correlational, or proportion-of-variance-accounted-for, terms.  As is well known, any F-ratio, even those obtained in classical analysis of variance designs, can be expressed in a metric that has a known upper and lower bound.  The metric most useful in this regard is either R² or 0^{2 (}Eta-square).  In the general linear model set up for regression and/or ANOVA problems, it is known that the SS_h = SS_total(R²), and that the SS_e = SS_total(1-R²).  Hence the ratio SS_h/SS_e can be written as R²/(1-R²) and Equation 3 can be rewritten as,

The same logic applies to equivalent values of 0²--the concepts 0² and R² are identical for our purposes.

     For purposes of power analysis, the first term to the right of the equal sign defines the effect size and the second term involving the ratio of the degrees of freedom defines the size of the study.  Most power tables (Cohen, 1988; Cohen, Cohen, West & Aiken, 2003) require that one have an estimate of the noncentrality parameter (say,  8) associated with the F-test.  This estimate of  8 based on Equation 4 is,

Notice that the noncentrality parameter is closely related to F, such that 8 = F(df_h ).  There are many noncentral distributions of F, one for every value of df_h.  To use the L-tables of Cohen & Cohen (1983), one estimates the value of  8, enters the table at the anticipated df_h, and reads the associated power value. 

     The power calculator of the top of this page performs the power calculations given an entry for R² or 0², the degrees of freedom of the model to be analyzed (df_h and df_e), and the desired alpha level (the alpha level also affects the power of an analysis; the lower the alpha, the lower the power, all other things being equal).  The program at the top of this page is convenient in that one can easily vary any or all of the four required pieces of the puzzle and investigate the statistical power of the analysis under different conditions.  Virtually all common effect sizes, and or test statistics,  found in the behavioral sciences can be converted to R² equivalents.  Rosenthal (1995) presents a thorough discussion of these conversions.  Those that cannot be expressed in R² equivalents (e.g., logistic regression) require different computational algorithms for estimating statistical power (see for example, Fleiss, 1995).

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Some Worked Examples

A Bivariate Correlation/Regression Problem

(1)   Consider a situation in which a study is to be run in which the Pearson Product-Moment Correlation will be the measure of association employed to assess the relationship between Y and X, and the t-test will be the test statistic.  Setting alpha = .05, anticipating that a correlation in the neighborhood of .30 will be found, a starting guess of a sample size of N = 50 will be used.  Entering r² = .09, df_h = 1, df_e = N - q_f - 1 = 50-1-1 = 48, the estimated power is .61.  Increasing the sample size to 100 (df_e = 98) yields an estimated power of .89, certainly an acceptable value.  It does not "guarantee" a significant result, but if the null hypothesis is false to the degree you estimated (r = .30), you will have 89 chances out of 100 of rejecting the null hypothesis with a sample size of 100--your power is 89%. 

A Two-Group Comparison Problem

(1)  An investigator wishes to test the difference between the means of two groups, one of which receives a treatment and the second of which is a classic control group.  The outcome measure is one of IQ with a population standard deviation of 15.  The investigator guesses that a 10 point difference between groups will be found.  Hence, the first part of the t-test formula is (Mean1 - Mean2)/S_common = 110-100/15 = 10/15 = .67.  The estimate of effect size suggests that the investigator expects a difference of 2/3's of a standard deviation.  The conversion of this effect size (Cohen's d) to an R² equivalent is found by computing r² = d²/(d² + 4) = .143.  Assuming two groups of equal size at n₁ = n₂ = 20, the df_e for the problem is N-q_f-1 = 40-1-1=38.  Setting alpha = .05 and using the power calculator, we find that the anticipated power is .73; the probability of rejecting a presumably false null hypothesis is approximately 7 in 10. 

(2)  A two-group study is planned which is quite similar to two published research studies that have appeared in the literature.  The first study reported a t-test of 2.15 on 30 degrees of freedom, the second study reported an F-test of 3.11 on 1 and 57 degrees of freedom.  A reasonable estimate of the effect size would be the average of the effect sizes from these two studies.  R² equivalents of R² = F(df_h)/[F(df_h)+df_e] from the second study = .052, and R² = t²/[t²+df^e] from the first study = .134, with average R² = .093.  The author plans to perform several tests and wishes to be statistically conservative and adopts an alpha = 01. Fifty subjects are available for the upcoming experiment (df_e = 48).  Hence the estimated statistical power under these conditions is .56.  The approximate 50/50 chance of rejecting the null hypothesis suggests that the author needs more subjects, should do fewer tests and raise the alpha level accordingly, or find a way (experimentally) to sharpen the design (induce better controls, reduce within group heterogeneity by blocking or covariance analysis).

A Full Model Multiple Regression Problem

(1)  A full model regression analysis problem requires estimates of the full model R² based on the q_f predictor variables in the model, the anticipated sample size, and the hypothesis degrees of freedom (the number of predictor vectors).  Assume that we anticipate an R² of about .10 on q_f = 6 predictors in a study with 150 participants.  Hence we estimate R² = .10, df_h = 6, df_e = 143.  The estimated power at alpha = .05 is .87.  A power of 87% would be more than acceptable to most investigators. 

A Semi-Partial Regression Problem

  Semi-partial analysis employs a test statistic (t or F) that is slightly differently constructed from the F-test for a full model analysis.  The F-test for the semipartial R² can best be seen as a test of a difference between a full and a restricted model.  The restriction placed on the model is specified by the hypothesis and the numerator of the F-test is actually the squared semi-partial R².  The semipartial R² can represent the unique contribution of either a single variable over and above the remaining variables in the model, or it can represent the unique contribution of a set of variables (0 < set < q_f) over and above the remaining variables in the model.  The appropriate expression for the F-test on the semipartial is [(R²_full - R²_restricted)/(1-R²_full)] x [df_e/df_h].  Notice that the denominator reflects a SS_error that is based on 1-R²_full, that is, the magnitude of the error term is reduced by all the variables, not just those that are involved in the semipartial R².  The numerator of the expression is the definition of a squared semipartial correlation coefficient.    

  Estimates for the semipartial correlation problem must be obtained by similar means as are employed in other problems.  The problem is more difficult to estimate since one must not only estimate the effect size for the full model R², but must estimate the effect size of the semipartial R² as well.  Consider this example.  An investigator wishes to estimate the power of a semipartial R² based on a set of 2 predictors that are embeded in a full model that has 5 predictors total.  Estimates are found (by experience, literature, pilot studies) that suggest the following values:  R²_full = .15, number of predictors in the full model 5, number of predictors in the restricted model (without the 2 predictors to be partialled) is threrefore 3.  The hypothesis degrees of freedom is 5-3=2, and it is anticipated that the semipartial R² will be approximately .05.  A sample size of 100 is anticipated.  The semipartial power subroutine applied to this problem predicts a power of .56 on 2 and 94 degrees of freedom.  Raising the sample size to 150 increases the power to .75 on 2 and 144 degrees of freedom.  To achieve a power of approximately 80% for this semipartial R², one would need approximately 165 observations.  To calculate power analyses on semipartial models, click here --> Semipartial Power Module

A One-Way Anova (GLM) Problem

   The one way anova problem from a linear model perspective can be construed as a problem in multiple regression analysis based on g-1 coded predictor vectors that represent group membership (g = the number of groups or levels of the factor).  No matter the what coding system is used (see Cohen, Cohen, West & Aiken for details of various coding systems) the Anova model contains the possibility for a proportion-of-variance-accounted-for interpretation of the analysis depending on the proportion of the total sum of squares that can be attributed to the model or group mean differences, that is, SS_model/SS_total.  In most Anova references this value is denoted as eta-squared (0²).  Calculation of the a-priori power of a single classification Anova model requires that you estimate the number of groups in the factor (df_effect = g-1), the sample size (and hence the df_error = n-g-1), and the anticipated value of 0².  Correspondence between 0² and other means of characterizing the F-test from the Anova, such as the variance among the group means, can be found in Cohen (1988).

   Assume that we wish to conduct a 4-group, one-way Anova and anticipate that will have 15 participants per group (N=60) and that we anticipate further that value of 0² will be on the order of .07.  Entering the power function above with df_h = 3, df_e = 55, and 0² = .07, we find the estimated power of the analysis to be .38.  To have only a slightly better than 1 in 3 chance of rejecting the null hypothesis appears low.  Adjusting the sample size estimate of the power function by trial and error eventually reveals that with an 0² of  .07 in a 4-group experiment, we will need at least 145 participants to achieve about 80% power for the analysis.  Preferably the subjects will be approximately equally distributed among the 4 groups, as unequal numbers of observations among the groups can lower the power of the analysis (Cohen, 1988).  

A Factorial Anova (GLM) Problem

   Factorial Anova designs have many of the same features that bear upon power analysis as do both single factor Anova and multiple regression designs.  The first order of business is to write down the design, the degrees of freedom for all elements in the design (i.e., effects and error), an finally to estimate a value of 0² for each effect in the design.  The value of 0² for each effect in the model must be estimated as a partial 0², which is defined by the ratio of the SS_effect/(SS_effect + SS_error).  Note that the partial  0² is not the semipartial R² discussed in the context of multiple regression.  One can estimate the semipartial 0² as well, but the computations are more cumbersome, and the F-tests on the semipartial and partial 0² are identical (Cohen, Cohen, West & Aiken, 2003).  Assume that we wish to conduct a power analysis for a 2 x 3 x 3 factorial Anova with three main effects (say, A, B, and C), three two-factor interactions (say, AB, AC, and BC), and one three-factor interaction (say, ABC).  Assume further that the values of 0² for the main effects are .06, .03, and .09, that the value of 0² for the two-way interactions are .05, .04, and .03, and that the value of 0² for the three-way interaction is .015.  Interaction effects are notoriously weaker than main effects (Cohen, 1988; Cohen, Cohen, West & Aiken, 2003; Venter & Maxwell, 2000) because of increased complexity of the construct, multicollinearity with the main effects that make up the product vector, and lower reliability of product vectors.  For this power investigation the sample size (N) is varied from 100 to 300, and sample sizes required  for each effect to reach an estimated power of 80%, are presented in Table 1.  Error degrees of freedom are denoted by the value N - Edf_h -1, where Edf_h are the degrees of freedom for all main and interaction effects in the model .

Table 1

   The decision regarding sample size will be a trade-off between maximizing power for a robust theoretical test of a given effect and the realistic limits of the sample sizes available under a given set of circumstances.  The power analysis investigation of this design suggests that for the main effects and two-way interactions 200 observations would probably suffice, but the three-way interaction would not have much chance of seeing the null hypothesis rejected.  To achieve 80% power with respect to the three-way interaction one would need about 600+ subjects in the study.  If the three-way interaction is a necessity for testing a theoretical concept, then you don't have much choice but to round up the 600 observations or to abandon the test.  Testing the three-way interaction with 200 subjects is not an adequate test of the null.  If you consider an n² of .015 to be clinically or practically meaningful and worth detecting if it exists at that level, you are going to need in excess of 600 observations to have an 80% chance of declaring it to be a reliable (non chance) phenomenon.  On the other hand, if you consider the 3-way interaction to be exploratory and if you do not find it to be significant, then you must come to one of two conclusions:  either it is so small as to be clinically irrelevant, or you don't have sufficient power to detect it.  Such are the dilemmas of using power analysis in the experimental design process.

A Multivariate Multiple Regression Analysis and Canonical Correlation

   Estimating power in the multivariate case is considerably more difficult than estimating power in the univariate case, mainly because the estimates of effect size and measures of strength of association are more complicated and more difficult to obtain.  Nonetheless, the same general principles apply and the computational function above is perfectly suitable for estimating power in this case if you can provide an estimate of effect size and the size of the experiment.  Ultimately what is required is a value of multivariate R², multivariate degrees of freedom for hypothesis and error, and a specification of the Type I error rate (alpha).  The Muller & Peterson (1984), Haase (1991), Cohen, Cohen, West & Aiken (2003, Chapter 12), Cohen (1988, Chapter 10) references present a relatively full account of obtaining the necessary estimates of R²_multivariate, df_hm, and df_em.  The process of performing multivariate power analysis is further complicated by the fact that there are no less than four different test statistics in common usage (and therefore, four different estimates of R²_multivariate, df_hm, and df_em).  Given these additional complexities and once a test statistic has been chosen (e.g., Wilks 7, Hotelling's Trace I, Pillai's Trace V, or Roy's GCR 2), routine computations of power based on the F-test approximations of any of the four multivariate test statistics can be obtained.  Since the first three test statistics are functions of the underlying eigenvalues (and canonical correaltions), then each of the test statistics is a test of the associated canonical correlation model.  Roy's GCR test is a test of the maximum canonical correlation. 

   As an example, assume that we anticipate the following 5 x 5 correlation matrix among the N subjects in a study:

    Consider a problem in which two dependent variables (say, Y₁ and Y₂) are to be predicted from a collection of three predictors (say, X₁, X₂, and X₃).  The necessary quantities for the solution to this estimated multivariate multiple regression problem depend on the hypothesis to be tested.  As in univariate analysis, several hypotheses are possible, including a test of the full model regression, a test of any partialled subset of of one or more predictor variables, or more complex hypothesis tests.  The computations to obtain the necessary multivariate quantities are tedious, but can be obtained by either SPSS MANOVA with matrix input (Hull & Nie, 1981, Harris, 2001) or other specialized programs (Haase, 1993 [download SPSS syntax], 2000 [download SPSS syntax]).  These programs render the current module immaterial as they compute all the necessary output plus the estimated power of the analysis for both multivariate tests and univariate follow up tests.  Nonetheless, we continue with the example from summary data.  The solution to the canonical correlation problem of the Y = f(X) problem cited above with an N = 55 yields a Pillai's Trace V = .230, and an estimated R²_multivariate = .115.  With p = 2 criterion variables and q_f = 3 predictor variables, the degrees of freedom for an approximate F-test of this relationship would be df_h = 6 and df_e = 102.  Entry of these values of (6, 102) into the power module above indicates that an error degrees of freedom of 102 would be required for a power of approximately 80% given full model multivariate regression based on this pattern of relationships documented in the correlation matrix.  For Pillai's Trace, the df_e = (N-q_f - 1 + s -p),  where s = minimum[p, q_f] = 2 for this problem.  Hence solving for N reveals that approximately 55 observations (cases, subjects, participants, units, etc) would be required to obtain a power of 80% with this specific configuration of correlations at alpha = .05. At a more stringent alpha of .01 with the same sample size of 55 and df_e = 102, the anticipated power would be approximately 57%.  An  increase in sample size to 100 would raise this power estimate at alpha = .01 to 92%.  Keep in mind that there is no necessary relationship between the multivariate and univariate powers for any given multivariate problem.  If the univariate tests of are of equal theoretical importance, their pfowers should be estimated separately.

   One must take great care to insure that the estimates of the correlations in the supermatrix R_YX are consistent and that they satisfy the triangular inequality requirement of correlation coefficients (see Tabachnick & Fidell, 2001; pp. 64-65).  That is, if the correlation between variables 1 and 2 is defined at a certain value (say, .70), and the correlation between variables 1 and 3 is defined at a fixed value (say, .50), then there are constraints placed on the correlation between variables 2 and 3; this third correlation cannot lie outside certain limits.  The danger of using pairwise deletion procedures allowed by most statistical packages (such as SPSS, SAS, SYSTAT, etc) is that estimates obtained on different sub samples can be inconsistent.  Pulling estimates from the published literature with different samples can also produce inconsistent estimates, as can producing estimates from one's experience and good guesses.  Be cautious in this area.  The best solution is to conduct a small pilot study to obtain estimates of the correlations which must, by definition, be consistent and to then use those estimates in a power analysis to settle on a sample size.  

A Multivariate Analysis of Variance Problem

A Coming attraction!!  Stay tuned.

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