Example 41.2 Two-Way ANOVA with Covariate

Suppose you can enhance the planned study discussed in Example 41.1 in two ways:

• incorporate results from races at two different altitudes ("high" and "low")

• measure the body mass index of each runner before the race

This is equivalent to adding a second fixed effect and a continuous covariate to your model.

Since lactic acid buildup is more pronounced at higher altitudes, you will include altitude as a factor in the model along with fluid, extending the one-way ANOVA to a two-way ANOVA. In doing so, you expect to lower the residual standard deviation from about 3.75 to 3.5 (in addition to generalizing the study results). You assume there is negligible interaction between fluid and altitude and plan to use a main-effects-only model. You conjecture that the mean lactic acid buildup follows Table 41.9.

By including a measurement of body mass index as a covariate in the study, you hope to further reduce the error variability. The extent of this reduction in variability is commonly expressed in two alternative ways: (1) the correlation between the covariates and the response or (2) the proportional reduction in total incurred by the covariates. You prefer the former and guess that the correlation between body mass index and lactic acid buildup is between 0.2 and 0.3. You specify these estimates with the NCOVARIATES= and CORRXY= options in the POWER statement. The covariate is not included in the MODEL statement.

You are interested in the same four fluid comparisons as in Example 41.1, shown in Table 41.8, except this time you want to marginalize over the effect of altitude.

For each of these contrasts, you want to determine the sample size required to achieve a power of 0.9 to detect an effect with magnitude according to Table 41.9. You are not yet attempting to choose a single sample size for the study, but rather checking the range of sample sizes needed by individual contrasts. You plan to test each contrast at . You will provide twice as many runners with water as with any of the electrolytes, and you predict that you can study approximately two-thirds as many runners at high altitude than at low altitude. The resulting planned sample size weighting scheme is shown in Table 41.10. Since the scheme is only approximate, you use the NFRACTIONAL option in the POWER statement to disable the rounding of sample sizes up to integers satisfying the weights exactly.

First, you create the exemplary data set to specify means and weights for the design profiles:

   data Fluids2;
      input Altitude $ Fluid $ LacticAcid CellWgt;
      datalines;
            High       Water      36.9       4
            High       EZD1       35.0       2
            High       EZD2       31.5       2
            High       LZ1        30         2
            High       LZ2        27.1       2
            Low        Water      34.3       6
            Low        EZD1       32.4       3
            Low        EZD2       28.9       3
            Low        LZ1        27         3
            Low        LZ2        24.7       3
      ;
   run;

The variables Altitude, Fluid, and LacticAcid specify the factors and cell means in Table 41.9. The variable CellWgt contains the sample size allocation weights in Table 41.10.

Use the DATA= option in the PROC GLMPOWER statement to specify Fluids2 as the exemplary data set. The following statements perform the sample size analysis:

   proc glmpower data=Fluids2;
      class Altitude Fluid;
      model LacticAcid = Altitude Fluid;
      weight CellWgt;
      contrast "Water vs. others" Fluid  -1 -1 -1 -1 4;
      contrast "EZD vs. LZ"       Fluid   1  1 -1 -1 0;
      contrast "EZD1 vs. EZD2"    Fluid   1 -1  0  0 0;
      contrast "LZ1 vs. LZ2"      Fluid   0  0  1 -1 0;
      power
         nfractional
         stddev      = 3.5
         ncovariates = 1
         corrxy      = 0.2 0.3 0
         alpha       = 0.025
         ntotal      = .
         power       = 0.9;
   run;

The CLASS statement identifies Altitude and Fluid as classification variables. The MODEL statement specifies the model, and the WEIGHT statement identifies CellWgt as the weight variable. The CONTRAST statement specifies the contrasts in Table 41.8. As in Example 41.1, the order of the contrast coefficients corresponds to the formatted class levels (EZD1, EZD2, LZ1, LZ2, Water). The POWER statement specifies total sample size as the result parameter and provides values for the other analysis parameters. The NCOVARIATES= option specifies the single covariate (body mass index), and the CORRXY= option specifies the two scenarios for its correlation with lactic acid buildup (0.2 and 0.3). Output 41.2.1 displays the results.