In Chapter 18 we introduced the concept of power as an important consideration in testing the null hypothesis. The purpose of this appendix is to describe statistical procedures for power analysis and estimation of sample size for studies using the *t*-test, analysis of variance, correlation, multiple regression, and chi-square for contingency tables. These procedures are based on the work of Cohen.^{1} For each procedure, formulas are provided, followed by specific examples of their use.

In power analysis we are concerned with five statistical elements: the significance criterion (*α*), the sample size (*n*), sample variance (*s*^{2}), effect size (ES), and power (1 – *β*). These elements are related in such a way that given any four, the fifth is readily determined.

*Effect size* is a measure of the magnitude of difference or correlation. The larger the observed effect, the more likely it will result in a significant statistical test (given a specific alpha level). An *effect size index* is a statistic that represents effect size using a standardized value that is universally applicable for all units of data, just as *t, F* and *r* are unit free. A different form of effect size index is used for each statistical procedure.

It is a simple process to calculate a sample effect size index following completion of a study. We know the sample size, and we can calculate the actual variance, means, correlations, or proportions in the data. This information can then be used to determine the degree of power achieved.

During planning stages of a study we use effect size to determine how many subjects will be needed. But because data are not yet available, the researcher must make an educated guess as to the expected effect size. This hypothesis is often based on previous research or pilot data, where studies can provide reasonable estimates for mean differences, correlations and variances. When such data are not available, the effect size estimate may be based on the researcher's opinion of a clinically meaningful difference; that is, the researcher can determine how large an effect would be important. For example, suppose we were interested in studying two treatments for improving shoulder range of motion in patients with adhesive capsulitis. We might say that the results of the treatments should differ by at least 20 degrees, or we would not consider the difference to be meaningful. Therefore, if we observed a difference this large, we would want it to be significant. This would be the effect size we would propose. Similarly, for a correlational study we could propose that a correlation of at least .60 would be important. These types of clinical judgments can then be used to guide the estimation of sample size.

When the researcher wants to establish the required sample size prior to data collection, and no clinical judgment or previous data provide a reasonable guide, Cohen proposes the use of conventional values, which are based on operational definitions for "small," "medium," and "large" effect sizes.^{1} Although these definitions are purely relative and somewhat intuitive, Cohen suggests that they represent reasonable estimates for planning purposes. Specific values for small, medium and large effects are proposed for each statistical procedure. Cohen emphasizes, however, that these descriptions are necessarily relative, and must be operationalized for a given research situation. A small effect size for tests of movement may be quite different from a small effect for psychological phenomena.

As a starting point then, a *small effect size* is considered small enough so that changes are not perceptible to the human eye, but not so small as to be minute. In new areas of inquiry, effect sizes are likely to be small because the phenomenon under study is typically not well understood and perhaps not under good experimental control. Many behavioral effects are likely to fall into this category, because of the influence of extraneous variables and the subtleties of human performance.

A *medium effect size* is conceived as large enough to be visible to the naked eye, so that one would be aware of the change in the course of normal observation.

A *large effect size* represents a great degree of separation, so that there is very little overlap between population distributions. Differences should be grossly observable. Large effect sizes are often seen in sociology, economics, and physiology, fields characterized by studies with large samples and good experimental control.

One way to conceptualize these definitions is to think of effect size in terms of variance. Using a simple framework involving two group means, the difference between means would be considered small if it is 20% of one standard deviation (assuming both groups have the same standard deviation). A medium effect would be equivalent to half a standard deviation, and a large effect would be 80% of a standard deviation. It is useful to think of effect size, then, as a ratio of the variance between groups relative to the variance within groups.

So what happens if the estimate of effect size is incorrect? What happens if we predict a large effect size and choose the appropriate sample, but the actual scores reveal a small effect, which turns out to be nonsignificant? Well, then we go back and determine the probability of a Type II error, and what level of power was actually achieved. This information can then be used for interpreting the study's results and in planning future studies. It is usually more prudent to be conservative in effect size estimates, so that a large enough sample will be recruited. If several analyses are planned for a given set of data (such as several regression equations or multiple analyses of variance), the sample size must be large enough to support the smallest hypothesized effect for the most complex analysis.^{2}

Tables are provided for power and sample size estimates at the end of this appendix. Power can be determined by knowing effect size and sample size, and sample size can be determined by knowing the expected effect size and the desired level of power. Tables are included for α = .05, for one- and two-tailed test where appropriate. Each statistical procedure requires its own set of tables. We have limited these tables to basic configurations for the *t*-test, analysis of variance, correlation, regression, and chi-square. The reader is referred to Cohen for additional tables.^{1}

Power analysis for the *t*-test is based on the effect size index, *d*, which expresses the difference between the two sample means in standard deviation units.

For the unpaired *t*-test with equal variances, the effect size index is calculated according to:

where *X̄*_{1} and *X̄*_{2} are the group means, and *s* is their common standard deviation. Assuming equality of variance, *s* can be the standard deviation from either group, or it can be their arithmetic average, the square root of the pooled variance, (see Equation 19.2).

If used after data analysis, the *d* index can also be computed using the calculated value of *t*:

With a nondirectional alternative hypothesis, only the absolute value of *d* is considered. With a directional hypothesis, the sign of *d* must correspond to the predicted direction.

When the assumption of homogeneity of variance is not met, the calculation of *d* is based on the *root mean square* (*s'*) of *s*_{1} and *s*_{2} as follows:

The value of *s'* is used in the denominator of Equation C.l.

When data are collected in a repeated measures design, calculation of *d* is based on paired scores. In this case, we first calculate *d'* using the means for the two test conditions and a common standard deviation:

We account for the fact that these values are correlated by adjusting *d* as follows:

where *r* is the correlation coefficient for the paired data.

When no estimate of *r* can be made, we can substitute the formula

where *d̄* is the mean of the difference scores, and *s _{d}* is the standard deviation of the difference scores. The value of

*d*to be used in the power tables is then determined by

When *d* cannot be computed directly, the following conventions can be used to assign value to the effect size index: small *d* = .20, medium *d* = .50, and large *d* = .80.

To determine the power achieved for a given sample size and effect size, we use Tables C.1.1 and C.1.2, found at the end of this appendix, for estimates at *α*_{1} and *α*_{2} = .05. Along the top of the table we locate the appropriate value of *d*, and down the side, the known sample size, *n*. For the unpaired test, sample size refers to the number of subjects in *each group* (assuming equal groups), not both groups combined. For the paired *t*-test, this is the number of subjects in the study. Power levels, in percentages, are given in the body of the table.

d | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

n | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | 1.00 | 1.20 | 1.40 |

8 | 07 | 10 | 13 | 19 | 25 | 31 | 38 | 46 | 61 | 74 | 85 |

9 | 07 | 11 | 15 | 20 | 27 | 34 | 41 | 50 | 66 | 79 | 88 |

10 | 08 | 11 | 16 | 22 | 29 | 36 | 45 | 53 | 70 | 83 | 91 |

11 | 08 | 12 | 17 | 23 | 31 | 39 | 48 | 57 | 74 | 86 | 94 |

12 | 08 | 12 | 18 | 25 | 33 | 41 | 51 | 60 | 77 | 89 | 96 |

13 | 08 | 13 | 18 | 26 | 34 | 44 | 54 | 63 | 80 | 91 | 97 |

14 | 08 | 13 | 19 | 27 | 36 | 46 | 57 | 66 | 83 | 93 | 98 |

15 | 08 | 13 | 20 | 28 | 38 | 48 | 59 | 69 | 85 | 94 | 98 |

16 | 09 | 14 | 21 | 30 | 40 | 51 | 62 | 72 | 87 | 95 | 99 |

17 | 09 | 14 | 22 | 31 | 42 | 53 | 64 | 74 | 89 | 96 | 99 |

18 | 09 | 15 | 22 | 32 | 43 | 55 | 66 | 76 | 90 | 97 | 99 |

19 | 09 | 15 | 23 | 33 | 45 | 57 | 68 | 78 | 92 | 98 | |

20 | 09 | 15 | 24 | 34 | 46 | 59 | 70 | 80 | 93 | 98 | |

30 | 10 | 19 | 31 | 46 | 61 | 74 | 85 | 92 | 99 | ||

40 | 11 | 22 | 38 | 55 | 72 | 84 | 93 | 97 | |||

50 | 12 | 26 | 44 | 63 | 80 | 91 | 97 | 99 | |||

100 | 17 | 41 | 66 | 88 | 97 | ||||||

200 | 26 | 64 | 91 | 99 |

d | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

n | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | 1.00 | 1.20 | 1.40 |

8 | 05 | 07 | 09 | 11 | 15 | 20 | 25 | 31 | 46 | 60 | 73 |

9 | 05 | 07 | 09 | 12 | 16 | 22 | 28 | 35 | 51 | 65 | 79 |

10 | 06 | 07 | 10 | 13 | 18 | 24 | 31 | 39 | 56 | 71 | 84 |

11 | 06 | 07 | 10 | 14 | 20 | 26 | 34 | 43 | 61 | 76 | 67 |

12 | 06 | 08 | 11 | 15 | 21 | 28 | 37 | 46 | 65 | 80 | 90 |

13 | 06 | 08 | 11 | 16 | 23 | 31 | 40 | 50 | 69 | 83 | 93 |

14 | 06 | 08 | 12 | 17 | 25 | 33 | 43 | 53 | 72 | 86 | 94 |

15 | 06 | 08 | 12 | 18 | 26 | 35 | 45 | 56 | 75 | 88 | 96 |

16 | 06 | 08 | 13 | 19 | 28 | 37 | 48 | 59 | 78 | 90 | 97 |

17 | 06 | 09 | 13 | 20 | 29 | 39 | 51 | 62 | 80 | 92 | 98 |

18 | 06 | 09 | 14 | 21 | 31 | 41 | 53 | 64 | 83 | 94 | 98 |

19 | 06 | 09 | 15 | 22 | 32 | 43 | 55 | 67 | 85 | 95 | 99 |

20 | 06 | 09 | 15 | 23 | 33 | 45 | 58 | 69 | 87 | 96 | 99 |

30 | 07 | 12 | 21 | 33 | 47 | 63 | 76 | 86 | 97 | ||

40 | 07 | 14 | 26 | 42 | 60 | 75 | 87 | 94 | 99 | ||

50 | 08 | 17 | 32 | 50 | 70 | 84 | 93 | 98 | |||

100 | 11 | 29 | 56 | 80 | 94 | 99 | |||||

200 | 17 | 51 | 85 | 98 |

When sample sizes are different for the two groups being compared, the *harmonic mean* of the two sample sizes, *n'*, is computed:

The value of *n'* is then used to locate *n* in the power table.

Table C.2 is used to determine the sample size needed for the *t*-test to achieve a desired level of power for one- and two-tailed tests at *α* = .05 and .01. For each subtable, the value of *d* is located across the top, and the desired power is given at the left. Power levels are listed for .70, .80, and .90. The sample sizes found in the body of the table represent the number of subjects required *in each group*, or the total number of subjects for paired observations.

d | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Power | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | 1.00 | 1.20 | 1.40 |

α_{1} = .05 | |||||||||||

.70 | 942 | 236 | 105 | 60 | 38 | 27 | 20 | 15 | 10 | 7 | 6 |

.80 | 1237 | 310 | 138 | 78 | 50 | 35 | 26 | 20 | 13 | 9 | 7 |

.90 | 1713 | 429 | 191 | 108 | 69 | 48 | 36 | 27 | 18 | 13 | 10 |

α_{2} = .05 | |||||||||||

.70 | 1235 | 310 | 138 | 78 | 50 | 35 | 26 | 20 | 13 | 10 | 7 |

.80 | 1571 | 393 | 175 | 99 | 64 | 45 | 33 | 26 | 17 | 12 | 9 |

.90 | 2102 | 526 | 234 | 132 | 85 | 59 | 44 | 34 | 22 | 16 | 12 |

When the exact value of *d* is not provided in the table, an adequate approximation of *n* can be given by

where *n*_{.10} is the sample size given for *d* = .10 in Table C.2, and *d* is the exact calculated value of the effect size index.^{∗}

^{∗}When using an average value for *n* the power estimates will be underestimates; that is, the power value in the table will be slightly lower than the true power. This underestimate will be trivial when *n'* > 25. Note that when sample sizes and variances are both unequal, the estimates of power using these tables may be inaccurate.^{1}

Consider the data from Table 19.1 in Chapter 19. In that hypothetical study, we measured change in pinch strength in two groups, with *X̄*_{1} = 10.11, *X̄*_{2} = 5.45, and *s*^{2}_{p} = 14.695. A directional hypothesis was proposed. Therefore, using Equation C.1,

Alternatively, *t* = 2.718, with *n* = 10 per group. Therefore, using Equation C.2,

To determine the power achieved with this test, we refer to Table C.1.1 for *α*_{1} = .05 and *n* = 10. With *d* = 1.2, we achieve 83% power. If we use these values to determine sample size for 80% power, we refer to Table C.2 for *α*_{1} = .05, where we find that we would need 9 subjects per group.

Now suppose we are planning this study. We state a nondirectional hypothesis and propose that a difference of 5 pounds would be important. We guess that a standard deviation of 8.0 would be expected. Therefore, we estimate that *d* = 5/8 = .625. Referring to Table C.2 for *α*_{1} = .05 and *d* = .60, we would estimate that we will need 35 subjects *per group* (a total of 70) to achieve 80% power. If once the study is completed, we obtain an effect size index of 1.2 with *n* = 35, we would then have achieved more than 99% power (Table C.1.1).

Consider the data in Table 19.2 in Chapter 19 for change in pinch strength, where *X̄*_{1} = 10.80, *X̄*_{2} = 5.65 and s^{2}_{1} = 25.17, s^{2}_{2} = 4.89. Therefore, using Equations C.3 and C.l,

Because the samples are of unequal size, we compute the harmonic mean to determine *n'* using Equation C.8. With *n*_{1} = 10 and *n*_{2} = 15:

Using *n* = 12 in Table C.1.1, we find for *d* = 1.2 power is 89%, and for *d* = 1.4 power is 96%. We can, therefore, estimate that power is approximately 93% for *d* = 1.3.

To determine sample size requirements with *d* = 1.3 we can use Table C.2. For 80% power with *α*_{1} = .05, we would need between 9 and 7 subjects (between *d* = 1.2 and 1.4). To calculate the exact sample size for *d* = 1.3, we use Equation C.9:

These results should always be rounded up to the nearest whole number. We would need 9 subjects per group to achieve 80% power with this effect size.

Consider the data in Table 19.3 in Chapter 19 for paired data. We examined the angle of the pelvis with and without a lumbar pillow in a sample of 8 subjects. For the paired *t*-test, we found that *X̄*_{1} = 102.38, *X̄*_{2} = 99.00 and *s*_{1} = 7.41, *s*_{2} = 8.64. A nondirectional hypothesis was proposed. The analysis also showed that *r* = .86 for the paired scores. Therefore, we use Equations C.4 and C.5:

Alternatively, using *d̄* = −3.375 and *s _{d}* = 6.232, with Equations C.6 and C.7:

Note that the minus sign is ignored.

To determine power at *α*_{2} = .05, we use Table C.1.2. For *n* = 8 and *d* = .80 (rounded) we can see that we achieve 31% power. This study had a good effect size, but the sample size was quite small, resulting in low power. Using Table C.2, we find we would have needed 26 subjects to achieve 80% power.

For the analysis of variance (ANOVA) the effect size index, *f*, is defined by

where *SS*_{e} is the error sum of squares from the ANOVA summary table.^{†} For a one-way ANOVA, *SS*_{b} is the between-groups sum of squares. For a two-way ANOVA, *SS*_{b} can represent either an individual main effect or the interaction effect; that is, a separate effect size index can be computed for each effect.^{‡} This index can be applied to independent samples and repeated measures designs.

If we do not have access to the ANOVA summary, we can also calculate *f* using the following formula:

where *s*_{m} is the standard deviation of the group means around the grand mean, and *s* is the common standard deviation for each group. For planning purposes, to estimate *f*, researchers may be able to hypothesize values for group means and their common standard deviation, based on theory and previous research. With equal sample sizes, *s*_{m} is obtained by

where represents the deviation of each individual group mean from the grand mean and *k* is the number of groups.^{§} Equation C.12 will work for the between-groups effect in a one-way ANOVA and for the main effects in a factorial design.

For two-way interactions, the *f* index must account for the variability among the interaction means with reference to the main effects (*A* and *B*) and the grand mean as follows:

where is the individual cell mean, is the marginal mean for variable *A* and is the marginal mean for variable *B* for that cell, and is the grand mean for the sample. The term *df _{AB}* represents the degrees of freedom associated with the interaction term (

*A*– 1)(

*B*− 1).

When effect sizes cannot be estimated from existing data, conventional values are as follows: small *f* = .10, medium *f* = .25, large *f* = .40.^{||}

^{†}Some statistical programs report this effect size index as eta squared These indices are related:^{1}

^{‡}Alternatively, an effect size index can also be computed for the overall two-way ANOVA model, combining all between-groups effects. The term *SS*_{b} would then be the sum of all between-groups sums of squares (i.e., *SS _{A}*,

*SS*, and

_{B}*SS*).

_{A×B}^{§}If groups are not of equal size, the difference between each group mean and the grand mean must be weighted by the sample size, using where *n _{i}* is the number of subjects in each group and

*N*is the total sample size for all groups combined.

^{||}Using *η*^{2} these conventional effect sizes are equivalent to small *η*^{2} = .01, medium *η*^{2} = .06, large *η*^{2} = .14.

Power tables for the analysis of variance are arranged according to the degrees of freedom associated with each *F*-test (d*f*_{b}). In a one-way ANOVA, this is the between-groups effect. In a two-way ANOVA (or larger) these effects will include each main effect and an interaction effect. Tables C.3.1, C.3.2, C.3.3, C.3.4 give power estimates for different values of the effect size index, *f*, at *df*_{b} = 1, 2, 3, and 4 at *α* = .05. See Cohen for additional tables.^{1}

f | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

n | .05 | .10 | .15 | .20 | .25 | .30 | .35 | .40 | .50 | .60 | .70 | .80 |

5 | 05 | 06 | 07 | 08 | 11 | 13 | 16 | 20 | 38 | 29 | 50 | 61 |

6 | 05 | 06 | 07 | 09 | 12 | 15 | 20 | 34 | 35 | 47 | 60 | 71 |

7 | 05 | 06 | 08 | 10 | 14 | 18 | 23 | 28 | 41 | 55 | 68 | 79 |

8 | 05 | 06 | 08 | 11 | 15 | 20 | 26 | 32 | 47 | 62 | 75 | 85 |

9 | 05 | 07 | 09 | 12 | 17 | 22 | 29 | 36 | 52 | 68 | 30 | 89 |

10 | 05 | 07 | 09 | 13 | 18 | 25 | 32 | 40 | 57 | 73 | 65 | 93 |

11 | 05 | 07 | 10 | 14 | 20 | 27 | 35 | 44 | 62 | 77 | 88 | 95 |

12 | 05 | 07 | 10 | 15 | 22 | 29 | 38 | 47 | 66 | 81 | 91 | 97 |

13 | 05 | 07 | 11 | 16 | 23 | 32 | 41 | 51 | 70 | 84 | 93 | 98 |

14 | 05 | 08 | 11 | 17 | 25 | 34 | 44 | 54 | 73 | 87 | 95 | 98 |

15 | 06 | 08 | 12 | 18 | 26 | 36 | 47 | 57 | 76 | 89 | 96 | 99 |

16 | 06 | 08 | 12 | 19 | 28 | 38 | 49 | 60 | 79 | 91 | 97 | 99 |

17 | 06 | 08 | 13 | 20 | 30 | 40 | 52 | 63 | 82 | 93 | 98 | |

18 | 06 | 08 | 14 | 21 | 31 | 42 | 54 | 66 | 84 | 94 | 98 | |

19 | 06 | 09 | 14 | 22 | 33 | 44 | 57 | 68 | 86 | 95 | 99 | |

20 | 06 | 09 | 15 | 23 | 34 | 46 | 59 | 70 | 88 | 96 | 99 | |

30 | 06 | 11 | 21 | 34 | 49 | 64 | 77 | 87 | 97 | |||

40 | 07 | 14 | 27 | 43 | 61 | 77 | 88 | 95 | 99 | |||

50 | 07 | 16 | 32 | 52 | 71 | 85 | 94 | 98 | ||||

60 | 08 | 19 | 38 | 60 | 79 | 94 | 97 | 99 | ||||

80 | 09 | 24 | 48 | 72 | 89 | 97 | 99 | |||||

100 | 10 | 29 | 57 | 81 | 94 | 99 | ||||||

120 | 11 | 34 | 65 | 88 | 94 | |||||||

140 | 13 | 39 | 72 | 92 | 99 | |||||||

160 | 14 | 44 | 77 | 95 | 99 | |||||||

180 | 15 | 48 | 82 | 97 | ||||||||

200 | 16 | 52 | 86 | 98 |

f | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

n | .05 | .10 | .15 | .20 | .25 | .30 | .35 | .40 | .50 | .60 | .70 | .80 |

5 | 05 | 06 | 07 | 09 | 11 | 14 | 17 | 22 | 32 | 44 | 56 | 69 |

6 | 05 | 06 | 07 | 10 | 13 | 16 | 21 | 26 | 39 | 53 | 67 | 79 |

7 | 05 | 06 | 08 | 11 | 14 | 19 | 25 | 31 | 46 | 62 | 76 | 87 |

8 | 05 | 06 | 08 | 12 | 16 | 22 | 28 | 36 | 53 | 69 | 83 | 92 |

9 | 05 | 07 | 09 | 13 | 18 | 24 | 32 | 40 | 59 | 75 | 88 | 95 |

10 | 05 | 07 | 10 | 14 | 20 | 27 | 35 | 45 | 64 | 81 | 91 | 97 |

11 | 05 | 07 | 10 | 15 | 21 | 30 | 39 | 49 | 69 | 85 | 94 | 98 |

12 | 06 | 07 | 11 | 16 | 23 | 32 | 42 | 53 | 74 | 88 | 96 | 99 |

13 | 06 | 08 | 11 | 17 | 25 | 35 | 46 | 57 | 77 | 91 | 97 | 99 |

14 | 06 | 08 | 12 | 18 | 27 | 38 | 49 | 61 | 81 | 93 | 98 | |

15 | 06 | 08 | 13 | 20 | 29 | 40 | 52 | 64 | 84 | 95 | 99 | |

16 | 06 | 08 | 13 | 21 | 31 | 43 | 55 | 94 | 86 | 96 | 99 | |

17 | 06 | 09 | 14 | 22 | 33 | 45 | 58 | 70 | 89 | 97 | 99 | |

18 | 06 | 09 | 14 | 23 | 34 | 48 | 61 | 73 | 90 | 98 | ||

19 | 06 | 09 | 15 | 24 | 36 | 50 | 64 | 76 | 92 | 99 | ||

20 | 06 | 09 | 16 | 26 | 38 | 52 | 66 | 78 | 93 | 99 | ||

30 | 06 | 12 | 22 | 37 | 55 | 71 | 85 | 93 | 99 | |||

40 | 07 | 15 | 29 | 48 | 68 | 84 | 94 | 98 | ||||

50 | 08 | 18 | 36 | 58 | 79 | 92 | 98 | 99 | ||||

60 | 08 | 21 | 42 | 67 | 86 | 96 | 99 | |||||

80 | 09 | 27 | 54 | 80 | 94 | 99 | ||||||

100 | 11 | 32 | 64 | 88 | 98 | |||||||

120 | 12 | 38 | 73 | 94 | 99 | |||||||

140 | 14 | 44 | 79 | 97 | ||||||||

160 | 15 | 49 | 85 | 98 | ||||||||

180 | 16 | 54 | 89 | 99 | ||||||||

200 | 18 | 59 | 92 |

f | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

n | .05 | .10 | .15 | .20 | .25 | .30 | .35 | .40 | .50 | .60 | .70 | .80 |

5 | 05 | 06 | 07 | 09 | 12 | 15 | 19 | 24 | 36 | 50 | 64 | 76 |

6 | 05 | 06 | 08 | 10 | 13 | 18 | 23 | 29 | 44 | 60 | 75 | 86 |

7 | 05 | 06 | 08 | 11 | 15 | 21 | 27 | 35 | 52 | 69 | 83 | 92 |

8 | 05 | 07 | 09 | 12 | 17 | 24 | 31 | 40 | 59 | 77 | 89 | 96 |

9 | 05 | 07 | 09 | 14 | 19 | 27 | 36 | 46 | 66 | 82 | 93 | 98 |

10 | 05 | 07 | 10 | 15 | 21 | 30 | 40 | 51 | 71 | 87 | 96 | 99 |

11 | 06 | 07 | 11 | 16 | 24 | 33 | 44 | 55 | 76 | 91 | 97 | 99 |

12 | 06 | 08 | 11 | 17 | 26 | 36 | 48 | 60 | 81 | 93 | 98 | |

13 | 06 | 08 | 12 | 19 | 28 | 39 | 52 | 64 | 84 | 95 | 99 | |

14 | 06 | 08 | 13 | 20 | 30 | 42 | 55 | 68 | 87 | 97 | 99 | |

15 | 06 | 08 | 13 | 21 | 32 | 45 | 59 | 71 | 90 | 98 | ||

16 | 06 | 09 | 14 | 23 | 34 | 48 | 62 | 75 | 92 | 98 | ||

17 | 06 | 09 | 15 | 24 | 37 | 51 | 65 | 78 | 94 | 99 | ||

18 | 06 | 09 | 16 | 26 | 39 | 53 | 68 | 80 | 95 | 99 | ||

19 | 06 | 09 | 16 | 27 | 41 | 56 | 71 | 83 | 96 | 99 | ||

20 | 06 | 10 | 17 | 28 | 43 | 59 | 73 | 85 | 97 | |||

30 | 07 | 13 | 25 | 42 | 61 | 79 | 90 | 96 | 99 | |||

40 | 07 | 16 | 32 | 54 | 76 | 90 | 97 | 99 | ||||

50 | 08 | 19 | 40 | 65 | 85 | 96 | 99 | |||||

60 | 09 | 22 | 47 | 74 | 91 | 98 | ||||||

80 | 10 | 29 | 61 | 86 | 97 | |||||||

100 | 11 | 36 | 71 | 93 | 99 | |||||||

120 | 13 | 43 | 80 | 97 | ||||||||

140 | 14 | 49 | 86 | 99 | ||||||||

160 | 16 | 55 | 94 | 99 | ||||||||

180 | 18 | 61 | 94 | |||||||||

200 | 19 | 66 | 96 |

f | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

n | .05 | .10 | .15 | .20 | .25 | .30 | .35 | .40 | .50 | .60 | .70 | .80 |

5 | 05 | 06 | 07 | 09 | 12 | 16 | 21 | 26 | 40 | 55 | 70 | 83 |

6 | 05 | 06 | 08 | 10 | 14 | 19 | 25 | 32 | 49 | 66 | 81 | 91 |

7 | 05 | 06 | 09 | 12 | 16 | 22 | 30 | 39 | 58 | 79 | 88 | 96 |

8 | 05 | 07 | 09 | 13 | 19 | 26 | 35 | 45 | 65 | 83 | 93 | 98 |

9 | 05 | 07 | 10 | 14 | 21 | 29 | 40 | 51 | 72 | 88 | 96 | 99 |

10 | 06 | 07 | 10 | 16 | 23 | 33 | 44 | 56 | 78 | 92 | 98 | |

11 | 06 | 08 | 11 | 17 | 26 | 37 | 49 | 61 | 82 | 94 | 99 | |

12 | 06 | 08 | 12 | 19 | 28 | 40 | 53 | 66 | 86 | 96 | 99 | |

13 | 06 | 08 | 13 | 20 | 31 | 43 | 57 | 70 | 89 | 98 | ||

14 | 06 | 08 | 13 | 22 | 33 | 47 | 61 | 74 | 92 | 98 | ||

15 | 06 | 09 | 14 | 23 | 36 | 50 | 65 | 78 | 94 | 99 | ||

16 | 06 | 09 | 15 | 25 | 38 | 53 | 68 | 81 | 95 | 99 | ||

17 | 06 | 09 | 16 | 26 | 40 | 56 | 71 | 83 | 96 | |||

18 | 06 | 09 | 17 | 28 | 43 | 59 | 74 | 86 | 97 | |||

19 | 06 | 10 | 17 | 30 | 45 | 62 | 77 | 88 | 98 | |||

20 | 06 | 10 | 18 | 31 | 47 | 65 | 79 | 90 | 99 | |||

30 | 07 | 13 | 27 | 46 | 67 | 84 | 94 | 98 | ||||

40 | 07 | 17 | 36 | 60 | 81 | 94 | 99 | |||||

50 | 08 | 21 | 44 | 81 | 90 | 98 | ||||||

60 | 09 | 24 | 52 | 80 | 95 | 99 | ||||||

80 | 10 | 32 | 66 | 91 | 99 | |||||||

100 | 12 | 40 | 77 | 96 | ||||||||

120 | 13 | 47 | 85 | 99 | ||||||||

140 | 15 | 54 | 91 | 99 | ||||||||

160 | 17 | 61 | 94 | |||||||||

180 | 18 | 67 | 97 | |||||||||

200 | 20 | 72 | 98 |

Sample sizes can be found using Table C.4 for various levels of *α* and *df*_{b}. These tables are used in the same way as the tables for the *t*-test.

f | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Power | .05 | .10 | .15 | .20 | .25 | .30 | .35 | .40 | .50 | .60 | .70 | .80 |

df = 1_{b} | ||||||||||||

.70 | 1235 | 310 | 138 | 78 | 50 | 35 | 26 | 20 | 13 | 10 | 7 | 6 |

.80 | 1571 | 393 | 175 | 99 | 64 | 45 | 33 | 26 | 17 | 12 | 9 | 7 |

.90 | 2102 | 526 | 234 | 132 | 85 | 59 | 44 | 34 | 22 | 16 | 12 | 9 |

df = 2_{b} | ||||||||||||

.70 | 1028 | 258 | 115 | 65 | 42 | 29 | 22 | 17 | 11 | 8 | 6 | 5 |

.80 | 1286 | 322 | 144 | 81 | 52 | 36 | 27 | 21 | 14 | 10 | 8 | 6 |

.90 | 1682 | 421 | 188 | 106 | 68 | 48 | 35 | 27 | 18 | 13 | 10 | 8 |

df = 3_{b} | ||||||||||||

.70 | 881 | 221 | 99 | 56 | 36 | 25 | 19 | 15 | 10 | 7 | 6 | 5 |

.80 | 1096 | 274 | 123 | 69 | 45 | 31 | 23 | 18 | 12 | 9 | 7 | 5 |

.90 | 1415 | 354 | 158 | 89 | 58 | 40 | 30 | 23 | 15 | 11 | 8 | 7 |

df = 4_{b} | ||||||||||||

.70 | 776 | 195 | 87 | 49 | 32 | 22 | 17 | 13 | 9 | 6 | 5 | 4 |

.80 | 956 | 240 | 107 | 61 | 39 | 27 | 20 | 16 | 10 | 8 | 6 | 5 |

.90 | 1231 | 309 | 138 | 78 | 50 | 35 | 26 | 20 | 13 | 10 | 7 | 6 |

df = 5_{b} | ||||||||||||

.70 | 698 | 175 | 78 | 44 | 29 | 20 | 15 | 12 | 8 | 6 | 5 | 4 |

.80 | 856 | 215 | 96 | 54 | 35 | 25 | 18 | 14 | 9 | 7 | 5 | 4 |

.90 | 1098 | 275 | 123 | 69 | 45 | 31 | 23 | 18 | 12 | 9 | 7 | 5 |

df = 6_{b} | ||||||||||||

.70 | 638 | 160 | 72 | 41 | 26 | 18 | 14 | 11 | 7 | 5 | 4 | 4 |

.80 | 780 | 195 | 87 | 50 | 32 | 22 | 17 | 13 | 9 | 6 | 5 | 4 |

.90 | 995 | 250 | 112 | 63 | 41 | 29 | 21 | 16 | 11 | 8 | 6 | 5 |

df = 8_{b} | ||||||||||||

.70 | 548 | 138 | 61 | 35 | 23 | 16 | 12 | 9 | 6 | 5 | 4 | 3 |

.80 | 669 | 168 | 75 | 42 | 27 | 19 | 14 | 11 | 8 | 6 | 4 | 4 |

.90 | 848 | 213 | 95 | 54 | 35 | 24 | 18 | 14 | 9 | 7 | 5 | 4 |

df = 10_{b} | ||||||||||||

.70 | 488 | 123 | 55 | 31 | 20 | 14 | 11 | 8 | 6 | 4 | 3 | 3 |

.80 | 591 | 148 | 66 | 38 | 24 | 17 | 13 | 10 | 7 | 5 | 4 | 3 |

.90 | 747 | 187 | 84 | 48 | 31 | 22 | 16 | 13 | 8 | 6 | 5 | 4 |

To find *n* for a value of *f* that is not tabled, we use

where *n*_{.05} is the sample size for *f* = .05 at the desired level of power and *f* is the exact value of the effect size index.

Consider the data for a one-way analysis of variance presented in Table 20.1 in Chapter 20. In this study we examined the effect of different modalities on ROM in 44 patients with elbow tendinitis. Four groups were compared (*k* = 4). Using data from the ANOVA output summary table, we found that *SS*_{b} = 3158.09 and *SS*_{e} = 3541.64. Therefore, using Equation C.10,

Given conventional effect sizes for *f*, this is a large effect.

To determine the power achieved with this test, we refer to Table C.3.3 for *df*_{b} = 3. The sample size used in the table refers to the number of subjects *in each group*, in this case *n* = 11. In this example, the *f* index is larger than .80, which is the highest value listed in the table. If we look across the row for *n* = 11, for *f* = .80 the power is 99%. Therefore, we can expect that power for *f* = 0.94 is 100%. Note that we actually would not do a power analysis for this study, as it resulted in a significant *F*-test. We use it here for illustration only.

Suppose we did not know these results, but we wanted to plan this study to determine the needed sample size. We hypothesize that ROM change will be greatest for those using ice, slightly less for those using ultrasound, less for those using massage, and much less for those with no intervention. As an estimate, based on our experience, we guess that the means will be 50, 40, 30, and 20, respectively. Using the literature as a guide, we also estimate that the standard deviation will be 8.0. With these values, we estimate a grand mean of (50 + 40 + 30 + 20)/4 = 35. Therefore, we can estimate *f* using Equations C.12 and C.ll as follows:

We turn to Table C.4 (*df*_{b} = 3) to determine the sample size needed to achieve 80% power with this estimated effect size. Using *f* = .80 (the largest value), we find that we would need 5 subjects per group, or a total sample of 20 subjects. Based on conventional effect sizes, this is considered a large effect, achieving high power even with a relatively small sample.

In a two-way ANOVA, power can be determined for each of the main effects and interaction effects. Consider the data presented in Table 20.3 in Chapter 20, for a study comparing the effect of three types of stretch (*A*) in two knee positions (*B*) for increasing ankle range of motion. A total of 60 subjects were assigned to six unique treatment groups. This analysis resulted in a significant interaction as well as a significant main effect for Stretch, but no significant effect for Position.

**Main Effect.** To look at the main effect of Position (variable *B*), we estimate effect size using Equation C.10 as follows:

where *SS _{B}* is the sum of squares associated with Position (variable

*B*).

^{#}Based on conventional values, this would be considered an extremely small effect. To determine the power of this test with two levels, we use Table C.3.1 for

*df*= 1. Each level of Position includes 30 subjects. This test achieves 6% power. Therefore, we have a 94% chance that we committed a Type II error. However, we must also consider the fact that the difference between main effect means for the two positions was 0.36 degrees (see Figure 20.6); that is, the observed effect size was extremely small. Given that we are measuring range of motion, such an effect would not be considered important. Therefore, it is unlikely that we have truly committed a Type II error. It is more likely that these positions are not different.

_{B}How many subjects would we have needed to achieve 80% power with this effect? We use Table C.4 for *df*_{b} = 1. For *f* = .05 at 80% power, we would need 1,571 subjects *per group*. As this is surely unreasonable, we might reconsider the inclusion of this variable in our research hypothesis.

**Interaction Effect.** To determine power for the interaction effect (an illustration, as this effect was significant), we use Equation C.10 as follows:

We refer to Table C.3.2 for *df*_{b} = 2 (the degrees of freedom associated with the interaction effect). The effect size values do not go as high as 1.00, but for *n* = 10 at *f* = .80, power is 97%. Therefore, we know that we have achieved maximal power for this effect.

In planning this study, suppose we hypothesized that prolonged stretch would be most effective, and that the control group would not change. We also believe that greater ankle range of motion will be achieved when stretch is given with the knee extended. We project the following means:

Knee Position | ||||
---|---|---|---|---|

Stretch | Flexion B_{1} | Extension B_{2} | Marginal Means | |

Prolonged | A_{1} | 15 | 20 | X̄_{A1} = 17.5 |

Quick | A_{2} | 10 | 10 | X̄_{A2} = 10 |

Control | A_{3} | 0 | 0 | X̄_{A3} = 0 |

Marginal Means | X̄_{B1} = 8.3 | X̄_{B2} = 10 | X̄_{G} = 9.15 |

From the literature we estimate that the standard deviation in our data will be 5.0. Using these values, we compute *S*_{m(AB)} using Equation C.13 as follows:

Therefore, according to Equation C.10,

To determine sample size, we look at Table C.4 for *df*_{b} = 2, power = 80%, and *f* = .30. We would need 36 subjects *per group*, or a total of 72 subjects. If we were doing this type of planning for a two-way design, we would project sample sizes for each main effect and interaction, and use the largest sample size as our guideline.

^{#}For these examples, please do not confuse subscripts *A* and *B*, which represent values associated with factors *A* and *B*, and subscript *b* which indicates the between-groups effect.

Power analysis for correlations is based on the magnitude of association, or the correlation coefficient. Because the correlation coefficient is a unit-free index, the effect size index does not need to be adjusted, and is simply the value of *r*.

Cohen addresses the dilemma that often surfaces when interpreting values of *r*; that is, even small correlations are often considered meaningful.^{1} This is especially common in the behavioral and clinical sciences, where significant correlations will often be less than .60. Therefore, how does one conceptualize a "large" or "small" effect? This must be a relative frame of reference, based on knowledge of the literature and clinical hypotheses; that is, how much of the variance in clinical phenomena can we truly expect to predict?

Based on this understanding, Cohen hesitates to offer conventional effect sizes for *r*, but does suggest the following may be used when no other statistical rationale is obvious: small *r* = 1.0, medium *r* = .30, large *r* = .50.

Tables C.5.1 and C.5.2 can be used to estimate power for the Pearson and Spearman correlations for one- and two-tailed tests at *α* = .05. Table C.6 provides estimated sample sizes required to achieve various levels of power for the same significance levels. The values of *n* in the tables represent the number of paired observations.

r | |||||||||
---|---|---|---|---|---|---|---|---|---|

n | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 |

8 | 08 | 12 | 18 | 26 | 37 | 52 | 68 | 85 | 97 |

9 | 08 | 13 | 20 | 29 | 42 | 57 | 74 | 90 | 99 |

10 | 08 | 14 | 22 | 32 | 46 | 62 | 79 | 93 | 99 |

11 | 09 | 15 | 23 | 35 | 50 | 67 | 63 | 95 | |

12 | 09 | 15 | 25 | 38 | 54 | 71 | 87 | 97 | |

13 | 09 | 16 | 26 | 40 | 57 | 74 | 89 | 98 | |

14 | 10 | 17 | 28 | 43 | 60 | 78 | 91 | 98 | |

15 | 10 | 18 | 30 | 45 | 63 | 81 | 93 | 99 | |

20 | 11 | 22 | 37 | 56 | 75 | 90 | 98 | ||

30 | 13 | 28 | 50 | 72 | 90 | 98 | |||

40 | 15 | 35 | 60 | 83 | 96 | ||||

50 | 17 | 41 | 69 | 90 | 98 | ||||

60 | 19 | 45 | 76 | 94 | 99 | ||||

80 | 22 | 45 | 86 | 98 | |||||

100 | 26 | 64 | 82 | 99 | |||||

200 | 41 | 89 | |||||||

500 | 72 | ||||||||

1000 | 94 |

r | |||||||||
---|---|---|---|---|---|---|---|---|---|

n | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 |

8 | 06 | 07 | 11 | 16 | 25 | 37 | 54 | 75 | 94 |

9 | 06 | 08 | 12 | 19 | 29 | 43 | 62 | 82 | 97 |

10 | 06 | 08 | 13 | 21 | 33 | 49 | 68 | 87 | 98 |

11 | 06 | 09 | 14 | 23 | 36 | 54 | 73 | 91 | 99 |

12 | 06 | 09 | 16 | 26 | 40 | 58 | 78 | 93 | 99 |

13 | 06 | 10 | 17 | 28 | 44 | 63 | 82 | 95 | |

14 | 06 | 10 | 18 | 30 | 47 | 66 | 85 | 96 | |

15 | 06 | 11 | 19 | 32 | 50 | 70 | 68 | 98 | |

20 | 07 | 14 | 25 | 43 | 64 | 83 | 96 | ||

30 | 08 | 19 | 37 | 61 | 83 | 95 | |||

40 | 09 | 24 | 48 | 74 | 92 | 99 | |||

50 | 11 | 29 | 57 | 83 | 97 | ||||

60 | 12 | 34 | 65 | 90 | 99 | ||||

80 | 14 | 43 | 78 | 96 | |||||

100 | 17 | 52 | 86 | 99 | |||||

200 | 29 | 81 | 99 | ||||||

500 | 61 | 99 | |||||||

1000 | 89 |

r | |||||||||
---|---|---|---|---|---|---|---|---|---|

Power | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 |

α_{1} = .05 | |||||||||

.70 | 470 | 117 | 52 | 28 | 18 | 12 | 8 | 6 | 4 |

.80 | 617 | 153 | 68 | 37 | 22 | 15 | 10 | 7 | 5 |

.90 | 654 | 211 | 92 | 50 | 31 | 20 | 13 | 9 | 6 |

α_{2} = .05 | |||||||||

.70 | 616 | 153 | 67 | 37 | 23 | 15 | 10 | 7 | 5 |

.80 | 763 | 194 | 85 | 46 | 28 | 18 | 12 | 9 | 6 |

.90 | 1047 | 259 | 113 | 62 | 37 | 24 | 16 | 11 | 7 |

Refer to Table 23.2 in Chapter 23, showing the correlation of proximal and distal behaviors in a sample of 12 normal infants, with a resulting correlation of *r* = .37, which was not significant at *α*_{1} = .05. To determine the level of power achieved, we refer to Table C.5.1. With *r* = .40 and 12 subjects, we attained 38% power; that is, there is a 62% chance we committed a Type II error. To find how many subjects we needed for 80% power, we use Table C.6 (*α*_{1} = .05). We should have recruited a sample of 37 subjects. In planning a study, we simply hypothesize a meaningful value for *r* and use this value in Table C.6.

In regression analysis, a quantitative dependent variable (*Y*) is correlated with a set of independent variables (*X*_{1}, *X*_{2}, through *X _{k}*). As with correlation, the degree of association within the regression equation represents the effect size, in this case

*R*

^{2}. The independent variables may represent quantitative or categorical variables (see Chapters 24 and 27).

In determining power, however, we must convert *R*^{2} to another index that will account for both the number of subjects and the number of independent variables in the regression. This index, called lambda (λ)^{∗∗'} is calculated as follows:

Table C.7 is used to determine the power of the regression for *α* = .05. To use this table we must know three elements: (1) the number of independent variables, *k*, in the leftmost column, (2) the number of residual degrees of freedom, *df*_{res} in the analysis of variance of regression (equal to *N* – *k* – 1), in the second column, and (3) the value for λ, along the top. For values of λ that fall between the values in the table, power can be determined by linear interpolation. Four values for *df*_{res} are given: 20, 60, 120, and ∞. Although not strictly linear, for degrees of freedom that fall between these values, power can be estimated with reasonable accuracy.

λ | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

k | df_{res} | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 24 | 28 | 32 | 36 | 40 |

1 | 20 | 27 | 48 | 64 | 77 | 85 | 91 | 95 | 97 | 98 | 99 | |||||

60 | 29 | 50 | 67 | 79 | 88 | 92 | 96 | 98 | 99 | 99 | ||||||

120 | 29 | 51 | 68 | 80 | 88 | 93 | 96 | 98 | 99 | 99 | ||||||

∞ | 29 | 52 | 69 | 81 | 89 | 93 | 96 | 98 | 99 | 99 | ||||||

2 | 20 | 20 | 36 | 52 | 65 | 75 | 83 | 88 | 92 | 95 | 97 | 99 | ||||

60 | 22 | 40 | 56 | 69 | 79 | 87 | 91 | 95 | 97 | 98 | ||||||

120 | 22 | 41 | 57 | 71 | 80 | 87 | 92 | 95 | 97 | 98 | ||||||

∞ | 23 | 42 | 58 | 72 | 82 | 88 | 93 | 96 | 97 | 99 | ||||||

3 | 20 | 17 | 30 | 44 | 56 | 67 | 75 | 82 | 87 | 91 | 94 | 97 | 99 | |||

60 | 19 | 34 | 49 | 62 | 73 | 81 | 87 | 92 | 95 | 97 | 98 | |||||

120 | 19 | 35 | 50 | 64 | 75 | 83 | 89 | 93 | 95 | 97 | 99 | |||||

∞ | 19 | 36 | 52 | 65 | 76 | 84 | 90 | 93 | 96 | 98 | 99 | |||||

4 | 20 | 15 | 26 | 38 | 49 | 60 | 69 | 76 | 83 | 87 | 91 | 95 | 98 | 99 | ||

60 | 17 | 30 | 44 | 57 | 68 | 77 | 83 | 89 | 92 | 95 | 98 | 99 | ||||

120 | 17 | 31 | 46 | 58 | 70 | 78 | 85 | 90 | 93 | 96 | 98 | 99 | ||||

∞ | 17 | 32 | 47 | 60 | 72 | 80 | 87 | 91 | 94 | 96 | 99 | |||||

5 | 20 | 13 | 23 | 34 | 44 | 54 | 63 | 71 | 78 | 83 | 87 | 93 | 96 | 98 | 99 | |

60 | 15 | 27 | 40 | 52 | 63 | 72 | 80 | 86 | 90 | 93 | 97 | 99 | ||||

120 | 16 | 29 | 41 | 54 | 65 | 75 | 82 | 87 | 91 | 94 | 98 | 99 | ||||

∞ | 16 | 29 | 43 | 56 | 68 | 77 | 84 | 89 | 93 | 95 | 98 | 99 | ||||

10 | 20 | 09 | 16 | 23 | 30 | 37 | 44 | 51 | 58 | 64 | 70 | 79 | 86 | 91 | 94 | 96 |

60 | 10 | 20 | 30 | 39 | 48 | 56 | 65 | 72 | 78 | 83 | 90 | 95 | 97 | 99 | 99 | |

120 | 11 | 21 | 31 | 42 | 51 | 60 | 69 | 75 | 81 | 86 | 93 | 96 | 98 | 99 | ||

∞ | 12 | 21 | 32 | 43 | 54 | 64 | 72 | 79 | 85 | 89 | 94 | 98 | 99 | |||

15 | 20 | 08 | 12 | 17 | 22 | 27 | 33 | 39 | 44 | 50 | 55 | 65 | 74 | 81 | 86 | 90 |

60 | 09 | 15 | 22 | 30 | 38 | 46 | 54 | 61 | 67 | 73 | 83 | 89 | 94 | 96 | 98 | |

120 | 10 | 16 | 24 | 33 | 42 | 51 | 59 | 66 | 73 | 78 | 87 | 92 | 96 | 98 | 99 | |

∞ | 10 | 18 | 27 | 37 | 47 | 56 | 64 | 72 | 78 | 83 | 91 | 95 | 97 | 99 | 99 | |

20 | 20 | 08 | 11 | 14 | 18 | 22 | 26 | 31 | 36 | 40 | 45 | 54 | 63 | 70 | 77 | 82 |

60 | 08 | 13 | 19 | 25 | 31 | 38 | 45 | 52 | 58 | 64 | 75 | 83 | 89 | 93 | 96 | |

120 | 09 | 14 | 21 | 28 | 36 | 43 | 51 | 58 | 65 | 71 | 81 | 88 | 93 | 96 | 98 | |

∞ | 09 | 16 | 24 | 32 | 41 | 50 | 58 | 65 | 72 | 78 | 87 | 92 | 96 | 98 | 99 |

To determine sample size, we specify a level of power, the number of independent variables, and a projected *R*^{2}. We then use Table C.8 to determine a value for lambda and substitute that value in the formula:

Power | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

k | df_{res} | .25 | .50 | .60 | .67 | .70 | .75 | .80 | .85 | .90 | .95 | .99 |

1 | 20 | 1.9 | 4.1 | 5.3 | 6.2 | 6.7 | 7.5 | 8.5 | 9.7 | 11.4 | 14.1 | 20.1 |

60 | 1.7 | 3.9 | 4.9 | 5.8 | 6.2 | 7.0 | 7.9 | 9.1 | 10.6 | 13.2 | 18.7 | |

120 | 1.7 | 3.8 | 4.9 | 5.7 | 6.2 | 6.9 | 7.8 | 9.0 | 10.5 | 13.0 | 18.4 | |

∞ | 1.6 | 3.8 | 4.9 | 5.7 | 6.2 | 6.9 | 7.8 | 9.0 | 10.5 | 13.0 | 18.4 | |

2 | 20 | 2.6 | 5.7 | 7.1 | 8.2 | 8.9 | 9.9 | 11.1 | 12.6 | 14.6 | 17.9 | 24.9 |

60 | 2.3 | 5.1 | 6.4 | 7.4 | 8.0 | 8.9 | 10.0 | 11.3 | 13.2 | 16.1 | 22.4 | |

120 | 2.3 | 5.0 | 6.3 | 7.2 | 7.8 | 8.7 | 9.7 | 11.1 | 12.8 | 15.7 | 21.8 | |

∞ | 2.2 | 5.0 | 6.2 | 7.2 | 7.7 | 8.6 | 9.6 | 10.9 | 12.7 | 15.4 | 21.4 | |

3 | 20 | 3.2 | 6.9 | 8.6 | 9.9 | 10.6 | 11.8 | 13.2 | 14.9 | 17.2 | 20.9 | 28.7 |

60 | 2.8 | 6.0 | 7.5 | 8.6 | 9.3 | 10.3 | 11.5 | 13.0 | 15.0 | 18.3 | 25.1 | |

120 | 2.7 | 5.8 | 7.3 | 8.4 | 9.0 | 10.0 | 11.1 | 12.6 | 14.5 | 17.7 | 24.3 | |

∞ | 2.7 | 5.8 | 7.2 | 8.2 | 8.8 | 9.8 | 10.9 | 12.3 | 14.2 | 17.2 | 23.5 | |

4 | 20 | 3.8 | 8.0 | 9.9 | 11.4 | 12.2 | 13.5 | 15.0 | 16.9 | 19.5 | 23.5 | 32.1 |

60 | 3.3 | 6.8 | 8.5 | 9.7 | 10.4 | 11.5 | 12.8 | 14.4 | 16.6 | 20.1 | 27.4 | |

120 | 3.1 | 6.6 | 8.1 | 9.3 | 10.0 | 11.0 | 12.3 | 13.9 | 16.0 | 19.3 | 26.3 | |

∞ | 3.1 | 6.4 | 7.9 | 9.1 | 9.7 | 10.7 | 11.9 | 13.4 | 15.4 | 18.6 | 25.2 | |

5 | 20 | 4.4 | 9.0 | 11.1 | 12.7 | 13.6 | 15.0 | 16.7 | 18.8 | 21.6 | 26.0 | 35.2 |

60 | 3.7 | 7.5 | 9.3 | 10.6 | 11.3 | 12.6 | 14.0 | 15.7 | 18.0 | 21.7 | 29.4 | |

120 | 3.5 | 7.2 | 8.9 | 10.1 | 10.8 | 12.0 | 13.3 | 15.0 | 17.2 | 20.7 | 28.1 | |

∞ | 3.4 | 7.0 | 8.6 | 9.8 | 10.5 | 11.6 | 12.8 | 14.4 | 16.5 | 19.8 | 26.7 | |

10 | 20 | 6.9 | 13.7 | 16.7 | 18.9 | 20.1 | 22.1 | 24.4 | 27.3 | 31.0 | 37.0 | 49.2 |

60 | 5.3 | 10.5 | 12.8 | 14.5 | 15.4 | 17.0 | 18.7 | 20.9 | 23.8 | 28.3 | 37.7 | |

120 | 5.0 | 9.8 | 11.9 | 13.5 | 14.3 | 15.8 | 17.4 | 19.5 | 22.1 | 26.4 | 35.2 | |

∞ | 4.4 | 9.2 | 11.2 | 12.6 | 13.4 | 14.7 | 16.8 | 18.1 | 20.5 | 24.4 | 32.4 | |

15 | 20 | 9.2 | 18.0 | 21.8 | 24.6 | 26.1 | 28.7 | 31.6 | 35.1 | 39.8 | 47.1 | 62.2 |

60 | 6.7 | 13.1 | 15.8 | 17.8 | 18.9 | 20.7 | 22.8 | 25.3 | 28.7 | 33.9 | 44.7 | |

120 | 6.1 | 11.9 | 14.3 | 16.2 | 17.2 | 18.8 | 20.7 | 23.1 | 26.1 | 30.9 | 40.8 | |

∞ | 5.6 | 10.9 | 13.1 | 14.7 | 15.6 | 17.1 | 18.8 | 20.9 | 23.6 | 27.8 | 36.6 | |

20 | 20 | 11.4 | 22.2 | 26.8 | 30.1 | 32.0 | 35.0 | 38.5 | 42.7 | 48.3 | 57.0 | 74.8 |

60 | 8.1 | 15.4 | 18.5 | 20.8 | 22.1 | 24.1 | 26.5 | 29.4 | 33.2 | 39.1 | 51.2 | |

120 | 7.2 | 13.7 | 16.5 | 18.6 | 19.7 | 21.6 | 23.7 | 26.3 | 29.6 | 34.9 | 45.8 | |

∞ | 6.4 | 12.3 | 14.7 | 16.5 | 17.5 | 19.1 | 21.0 | 23.2 | 26.1 | 30.7 | 40.1 |

The obvious dilemma in this process is that finding a value for lambda requires estimating *df _{res}*, which is a function of sample size, which we are trying to determine! Therefore, the process becomes one of limited trial and error. We start with one value for lambda, determine the associated sample size, and then calculate

*df*for that sample (

_{res}*df*=

_{res}*N*–

*k*– 1). If the numbers do not correspond, we go back and choose a different value for

*df*and try again. One choice in the table will provide a reasonable estimate.

_{ies}^{∗∗}Lambda is the *noncentrality parameter* of the *F*-test. Some statistical programs, such as SPSS, will generate a value for the noncentrality parameter as part of a power analysis for an analysis of variance. This value by itself is not used to represent power, but is needed to obtain power estimates.

To illustrate power analysis for multiple regression, consider a hypothetical study involving five independent variables (*k* = 5) as predictors of hospital length of stay (LOS). Suppose our sample consists of 30 patients, and results show that *R ^{2}* = .20 (

*p*= .176). Because this is not significant, we want to determine the power of the test. First we calculate lambda for

*k*= 5,

*R*

^{2}= .20,

*df*= 24, and

_{ies}*N*= 30, using Equation C.15:

We refer to Table C.7. Using the closest values for *df*_{res} = 20 *k* = 5, and λ = 8, we find that our test achieved approximately 44% power, indicating a 56% probability of committing a Type II error.

Table C.7 also shows us how the number of independent variables included in the regression will influence the number of subjects needed. As *k* increases, we can see that power decreases for a given value of λ. Just to illustrate, look at *k* = 3 for λ = 14 at *df*_{res} = 20. Power is 82%. If we double the number of independent variables, *k* = 6, power decreases to 66%. With 10 independent variables, we are down to 51%. Researchers will often use stepwise regression or factor analysis to decrease the number of independent variables in a regression analysis, which clearly will have the effect of improving power.

Now let us suppose we are planning this study and we want to determine how many subjects would be needed to achieve 80% power in the analysis with five independent variables. A literature search suggests that the hypothesized effect will be *R ^{2}* = .40. We start by referring to Table C.8 to determine the value for lambda. Since we do not know how many subjects we need, we must first choose a trial value for

*df*

_{res}. Cohen suggests that, as the values of lambda do not vary greatly among the four choices for residual degrees of freedom, using a trial value for

*df*

_{res}= 120 will generally yield an

*N*of sufficient accuracy.

^{1}Starting there, for

*k*= 5 at 80% power, we find

*λ*= 13.3. Using Equation C.16, we determine

*N*as follows:

This projection tells us we would need 20 subjects to achieve 80% power.^{††} If we use this estimate, the residual degrees of freedom would then be *N* – *k* – 1 = 20 – 5 – 1 = 14. Obviously, a great disparity exists between this value and *df _{res}* = 120, which we used to calculate

*N*. This is the trial and error part. Now we return to Table C.8 and find λ = 16.7 for

*k*= 5 and 80% power at

*df*

_{res}= 20, which we guess will be closer to our required

*N*. This time we find

If we use a sample of 25 subjects, *df*_{res} will be 25 – 5 – 1 = 19, which corresponds with the tabled values, so we can be comfortable with the outcome. Note that in planning this study, we hypothesized a value for *R ^{2}* that is much higher than the value we actually obtained, and therefore, the sample size estimate would not have been adequate for finding a significant effect. Projections of sample size are only as good as the projected effect size.

^{††}We can demonstrate that using 120 degrees of freedom for the calculation will yield a value for *N* that is not much different from those that would be calculated using the other values. For instance, for *df*_{res} = 60, λ = 11.5, which would yield *N* = 36.42. For *df*_{res} = 20, λ = 13.2, which would yield *N* = 41.8. We can generally expect these sample sizes to vary by no more than 10 subjects. Cohen does provide a formula for obtaining a more exact value of *N* using an adjusted value for λ.^{1}

We can establish power for the chi-square test for goodness of fit tests as well as contingency tables. The effect size index is given the symbol *w*.

For a 2 × 2 contingency table,^{‡‡}

For a contingency table with more than two rows or columns,^{§§}

where *q* is the number of rows or columns, *whichever is smaller.*

Cohen offers values for conventional effect sizes: small *w* = .10, medium *w* = .30, large *w* = .50. He suggests, however, that these values be used with caution, as the value of *w* will vary with the number of rows, columns, and degrees of freedom in a set of data, even when the true degree of association is the same.

^{‡‡}Note that this value is identical to the phi coefficient, described in Chapters 23 and 25.

^{§§}Note that this value is related to Cramer's *V*, described in Chapter 25.

Tables C.9.1, C.9.2, C.9.3, C.9.4 provide power estimates for *α* = .05 for degrees of freedom = 1, 2, 3, and 4 associated with chi-square [(*R* − 1)(*C* − 1)]. To use the tables, we must specify the overall sample size (*N*) and the value of *w*. Table C.10 provides sample size estimates, based on a given *α* level and degrees of freedom. If the value for *w* is not given in the table, we can determine an exact *N* according to:

*where* *N*_{.10} is the required sample size for *w* = .10 for the desired power and degrees of freedom.

w | |||||||||
---|---|---|---|---|---|---|---|---|---|

N | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 |

25 | 08 | 17 | 32 | 52 | 70 | 85 | 94 | 98 | 99 |

30 | 08 | 19 | 38 | 59 | 78 | 91 | 97 | 99 | |

35 | 09 | 22 | 43 | 66 | 84 | 94 | 99 | ||

40 | 10 | 24 | 47 | 71 | 89 | 97 | 99 | ||

45 | 10 | 27 | 52 | 76 | 92 | 98 | |||

50 | 11 | 29 | 56 | 81 | 94 | 99 | |||

60 | 12 | 34 | 64 | 87 | 97 | ||||

70 | 13 | 39 | 71 | 92 | 99 | ||||

80 | 15 | 43 | 76 | 95 | 99 | ||||

90 | 16 | 47 | 81 | 97 | |||||

100 | 17 | 52 | 85 | 98 | |||||

120 | 19 | 59 | 91 | 99 | |||||

140 | 22 | 66 | 94 | ||||||

160 | 24 | 71 | 97 | ||||||

180 | 27 | 76 | 98 | ||||||

200 | 29 | 61 | 99 | ||||||

300 | 41 | 93 | |||||||

500 | 61 | 99 | |||||||

600 | 69 | ||||||||

700 | 75 | ||||||||

600 | 81 | ||||||||

900 | 85 | ||||||||

1000 | 89 |

w | |||||||||
---|---|---|---|---|---|---|---|---|---|

N | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 |

25 | 07 | 13 | 25 | 42 | 60 | 77 | 89 | 96 | 99 |

30 | 07 | 15 | 29 | 49 | 69 | 85 | 94 | 98 | |

35 | 08 | 17 | 34 | 55 | 76 | 90 | 97 | 99 | |

40 | 08 | 19 | 38 | 61 | 82 | 93 | 98 | ||

45 | 09 | 21 | 42 | 67 | 86 | 96 | 99 | ||

50 | 09 | 23 | 46 | 72 | 90 | 97 | |||

60 | 10 | 26 | 54 | 80 | 94 | 99 | |||

70 | 11 | 30 | 61 | 86 | 97 | ||||

80 | 12 | 34 | 67 | 90 | 99 | ||||

90 | 12 | 38 | 72 | 93 | 99 | ||||

100 | 13 | 42 | 77 | 96 | |||||

120 | 15 | 49 | 85 | 98 | |||||

140 | 17 | 55 | 90 | 99 | |||||

160 | 19 | 61 | 93 | ||||||

180 | 21 | 67 | 96 | ||||||

200 | 23 | 72 | 97 | ||||||

300 | 32 | 88 | |||||||

500 | 50 | 99 | |||||||

600 | 58 | ||||||||

700 | 66 | ||||||||

800 | 72 | ||||||||

900 | 77 | ||||||||

1000 | 82 |

w | |||||||||
---|---|---|---|---|---|---|---|---|---|

N | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 |

25 | 07 | 12 | 21 | 36 | 54 | 71 | 85 | 93 | 98 |

30 | 07 | 13 | 25 | 42 | 62 | 80 | 90 | 97 | 99 |

35 | 07 | 15 | 29 | 49 | 70 | 86 | 95 | 99 | |

40 | 07 | 16 | 32 | 55 | 76 | 90 | 97 | 99 | |

45 | 08 | 18 | 36 | 60 | 81 | 94 | 99 | ||

50 | 08 | 19 | 40 | 65 | 86 | 96 | 99 | ||

60 | 09 | 22 | 47 | 74 | 92 | 98 | |||

70 | 09 | 26 | 54 | 81 | 95 | 99 | |||

80 | 10 | 29 | 60 | 86 | 98 | ||||

90 | 11 | 32 | 66 | 90 | 99 | ||||

100 | 12 | 36 | 71 | 93 | 99 | ||||

120 | 13 | 42 | 80 | 97 | |||||

140 | 15 | 49 | 86 | 99 | |||||

160 | 16 | 55 | 90 | 99 | |||||

180 | 18 | 60 | 94 | ||||||

200 | 19 | 65 | 96 | ||||||

300 | 27 | 84 | |||||||

500 | 44 | 98 | |||||||

600 | 52 | 99 | |||||||

700 | 59 | ||||||||

800 | 65 | ||||||||

900 | 71 | ||||||||

1000 | 76 |

w | |||||||||
---|---|---|---|---|---|---|---|---|---|

N | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 |

25 | 06 | 11 | 19 | 32 | 50 | 66 | 81 | 91 | 97 |

30 | 07 | 12 | 22 | 38 | 57 | 75 | 88 | 96 | 99 |

35 | 07 | 13 | 26 | 44 | 65 | 82 | 93 | 98 | |

40 | 07 | 14 | 29 | 50 | 72 | 88 | 96 | 99 | |

45 | 07 | 16 | 32 | 55 | 77 | 92 | 98 | ||

50 | 06 | 17 | 36 | 60 | 82 | 94 | 99 | ||

60 | 08 | 20 | 43 | 70 | 89 | 98 | |||

70 | 09 | 23 | 49 | 77 | 94 | 99 | |||

80 | 09 | 26 | 55 | 83 | 96 | ||||

90 | 10 | 29 | 61 | 88 | 98 | ||||

100 | 11 | 32 | 66 | 91 | 99 | ||||

120 | 12 | 38 | 75 | 96 | |||||

140 | 13 | 44 | 82 | 98 | |||||

160 | 14 | 50 | 88 | 99 | |||||

160 | 16 | 55 | 92 | ||||||

200 | 17 | 60 | 94 | ||||||

300 | 24 | 80 | 99 | ||||||

500 | 40 | 96 | |||||||

600 | 47 | 99 | |||||||

700 | 54 | ||||||||

800 | 60 | ||||||||

900 | 66 | ||||||||

1000 | 72 |

w | |||||||||
---|---|---|---|---|---|---|---|---|---|

Power | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 |

df = 1 | |||||||||

.70 | 617 | 154 | 69 | 39 | 25 | 17 | 13 | 10 | 8 |

.80 | 785 | 196 | 87 | 49 | 31 | 22 | 16 | 12 | 10 |

.90 | 1051 | 263 | 117 | 66 | 42 | 29 | 21 | 16 | 13 |

df = 2 | |||||||||

.70 | 770 | 193 | 86 | 48 | 31 | 21 | 16 | 12 | 10 |

.80 | 964 | 241 | 107 | 60 | 39 | 27 | 20 | 15 | 12 |

.90 | 1265 | 316 | 141 | 79 | 51 | 35 | 26 | 20 | 16 |

df = 3 | |||||||||

.70 | 879 | 220 | 98 | 55 | 35 | 24 | 18 | 14 | 11 |

.80 | 1090 | 273 | 121 | 68 | 44 | 30 | 22 | 17 | 13 |

.90 | 1417 | 354 | 157 | 89 | 57 | 39 | 29 | 22 | 17 |

df = 4 | |||||||||

.70 | 968 | 242 | 108 | 61 | 39 | 27 | 20 | 15 | 12 |

.80 | 1194 | 298 | 133 | 75 | 48 | 33 | 24 | 19 | 15 |

.90 | 1540 | 385 | 171 | 96 | 62 | 43 | 31 | 24 | 19 |

df = 6 | |||||||||

.70 | 1114 | 279 | 124 | 70 | 45 | 31 | 23 | 17 | 14 |

.80 | 1362 | 341 | 151 | 85 | 54 | 38 | 28 | 21 | 17 |

.90 | 1742 | 435 | 194 | 109 | 70 | 48 | 36 | 27 | 22 |

df = 8 | |||||||||

.70 | 1235 | 309 | 137 | 77 | 49 | 34 | 25 | 19 | 15 |

.80 | 1502 | 376 | 167 | 94 | 60 | 42 | 31 | 23 | 19 |

.90 | 1908 | 477 | 212 | 119 | 76 | 53 | 39 | 30 | 24 |

df = 9 | |||||||||

.70 | 1289 | 322 | 143 | 81 | 52 | 36 | 26 | 20 | 16 |

.80 | 1565 | 391 | 174 | 98 | 63 | 43 | 32 | 24 | 19 |

.90 | 1983 | 496 | 220 | 124 | 79 | 55 | 40 | 31 | 24 |

df = 12 | |||||||||

.70 | 1435 | 359 | 159 | 90 | 57 | 40 | 29 | 22 | 18 |

.80 | 1734 | 433 | 193 | 108 | 69 | 48 | 35 | 27 | 21 |

.90 | 2183 | 546 | 243 | 136 | 87 | 61 | 45 | 34 | 27 |

To illustrate this application for a 2 × 2 contingency table, refer to the study described in Chapter 25, Table 25.3. This study examined the frequency of diabetic wound healing with a total contact cast (TCC) and a removable cast walker (RCW) (a 2 × 2 design). For these data, *χ*^{2} = 5.24 and *N* = 50. This study did result in a significant chi-square test, but we will illustrate the process using this data. We can apply these data to determine the probability of committing a Type II error using Equation C.17:

Using Table C.9.1, for 1 degree of freedom, this study achieves 56% power (at *w* = .30). We can estimate an associated probability of Type II error of 44%.

To determine how many subjects we would need to achieve 80% power, we refer to Table C.10. For *w =* .30, we need 87 subjects (substantially more than the original study used).

*Statistical Power Analysis for the Behavioral Sciences.*2d ed. Hillsdale, NJ: Lawrence Erlbaum Associates, 1988.

*Using SPSS for Windows: Analyzing and Understanding Data*(4th ed.). Upper Saddle River, NJ: Prentice-Hall, 2004.