Question

The experiment described in Example \(15.4\) also gave data on change in body fat mass for men ("Growth Hormone and Sex Steroid Administration in Healthy Aged Women and Men," Journal of the American Medical Association [2002]: 2282-2292). Each of 74 male subjects who were over age 65 was assigned at random to one of the following four treatments: (1) placebo "growth hormone" and placebo "steroid" (denoted by \(\mathrm{P}+\mathrm{P}),(2)\) placebo "growth hormone" and the steroid testosterone (denoted by \(\mathrm{P}+\mathrm{S}\) ), (3) growth hormone and placebo "steroid" (denoted by G + P), and (4) growth hormone and the steroid testosterone (denoted by \(\mathrm{G}+\mathrm{S}\) ). The accompanying table lists data on change in body fat mass over the 26-week period following the treatment that are consistent with summary quantities given in the article $$\begin{array}{rrrr} \text { Treatment } \quad \mathbf{P}+\mathbf{P} & \mathbf{P}+\mathbf{S} & \mathbf{G}+\mathbf{P} & \mathbf{G}+\mathbf{S} \\ \hline 0.3 & -3.7 & -3.8 & -5.0 \\ 0.4 & -1.0 & -3.2 & -5.0 \\ -1.7 & 0.2 & -4.9 & -3.0 \\ -0.5 & -2.3 & -5.2 & -2.6 \\ -2.1 & 1.5 & -2.2 & -6.2 \\ 1.3 & -1.4 & -3.5 & -7.0 \\ 0.8 & 1.2 & -4.4 & -4.5 \\ 1.5 & -2.5 & -0.8 & -4.2 \\ -1.2 & -3.3 & -1.8 & -5.2 \\ -0.2 & 0.2 & -4.0 & -6.2 \\ 1.7 & 0.6 & -1.9 & -4.0 \\ 1.2 & -0.7 & -3.0 & -3.9 \end{array}$$ $$\begin{array}{rrrrr} \text { Treatment } & \mathbf{P}+\mathbf{P} & \mathbf{P}+\mathbf{S} & \mathbf{G}+\mathbf{P} & \mathbf{G}+\mathbf{S} \\ \hline & 0.6 & -0.1 & -1.8 & -3.3 \\ & 0.4 & -3.1 & -2.9 & -5.7 \\ & -1.3 & 0.3 & -2.9 & -4.5 \\ & -0.2 & -0.5 & -2.9 & -4.3 \\ & 0.7 & -0.8 & -3.7 & -4.0 \\ & & -0.7 & & -4.2 \\ & & -0.9 & & -4.7 \\ & & -2.0 & & \\ & & -0.6 & & \\ n & 17 & 21 & 17 & 19 \\ \bar{x} & 0.100 & -0.933 & -3.112 & -4.605 \\ s & 1.139 & 1.443 & 1.178 & 1.122 \\ s^{2} & 1.297 & 2.082 & 1.388 & 1.259 \end{array}$$ Also, \(N=74\), grand total \(=-158.3\), and \(\overline{\bar{x}}=\frac{-158.3}{74}=\) \(-2.139 .\) Carry out an \(F\) test to see whether true mean change in body fat mass differs for the four treatments.

Short answer

The result of the F-test will show whether there is a statistically significant difference between the group means. A high F-score will suggest we reject the null hypothesis and conclude there are significant differences between the groups. A low F-score will suggest we retain the null hypothesis, concluding there are no significant differences between the groups. The actual F-score value will be dependent on the provided data.

Step-by-step solution

Calculate the sum of squares (SS)

First, calculate the sum of squares within groups (SSW) and the sum of squares between groups (SSB). SSW is the sum of the variances of each group multiplied by the number of members in each group minus 1. These can be found using the provided variances \(s^{2}\) and the respective number of members in each group \(n\). SSB is the sum of the squared differences between each group's mean and the overall mean, multiplied by the number of members in each group.

Calculate the degrees of freedom (df)

The degrees of freedom for SSW (dfW) and SSB (dfB) need to be calculated next. dfW is the total number of subjects minus the number of groups, and dfB is the number of groups minus 1.

Calculate the mean squares (MS)

Next, calculate the mean squares within groups (MSW) and the mean squares between groups (MSB). MSW is the SSW divided by dfW, and MSB is the SSB divided by dfB.

Calculate the F-statistic

The F statistic is calculated by dividing MSB by MSW.

Decision to Reject or Retain Null Hypothesis

Compare the F statistic to the critical value from the F distribution with dfB and dfW degrees of freedom. If the F statistic is greater than the critical value, then the null hypothesis that the different treatments result in the same mean change can be rejected. If the F statistic is less than or equal to the critical value, then there is not enough evidence to reject the null hypothesis.