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Chapter 15: Problem 4

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

Expert verified
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

01

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.
02

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.
03

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.
04

Calculate the F-statistic

The F statistic is calculated by dividing MSB by MSW.
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum of Squares - Understanding Variability
Sum of squares is a statistical tool used to measure the variability in data. When performing ANOVA (Analysis of Variance), we often look at how much of the total variation within our data can be attributed to different sources. In the context of the given exercise, we break down the variation into two parts:
  • Sum of Squares Within groups (SSW): This represents the variation due to differences within individual groups. In simple terms, it's the variation of data points within each treatment group around its own mean.
  • Sum of Squares Between groups (SSB): This is the variation due to the difference between the group means and the overall mean of all data. It tells us how much of the total variation can be explained by the differences in treatment effects.
To calculate SSW, using the variances within each group (\(s^2\)), multiply them by the number of subjects in each group minus 1. For SSB, consider the difference between each group mean and the overall mean, square these differences, and multiply by the number of subjects in each group.
By summing SSB and SSW, we can determine the total variation seen in the experiment.
Degrees of Freedom - A Measure of Independence
Degrees of freedom refer to the number of independent values that can vary in an analysis without breaking any constraints. For ANOVA, there are degrees of freedom associated with both SSB and SSW.
  • Degrees of Freedom Within (dfW): This is calculated as the total number of observations minus the number of groups. It signifies the number of independent comparisons possible within groups.
  • Degrees of Freedom Between (dfB): This is one less than the number of groups, reflecting the number of independent comparisons possible between groups.
In our exercise, with four treatment groups each having their own data points, dfB would be 3 (since there are four groups), and dfW would be the total number of subjects (74) minus the number of groups (4), giving us 70.
These degrees of freedom help determine the shapes of the sampling distributions used in hypothesis testing.
Mean Squares - Averaging the Variability
Mean squares are crucial in ANOVA as they provide an average measure of variability by dividing the sum of squares by their corresponding degrees of freedom. In simpler terms:
  • Mean Square Within (MSW): This is obtained by dividing the SSW by dfW. It measures the average variability inside each treatment group.
  • Mean Square Between (MSB): Calculate this by dividing the SSB by dfB. This value shows the average variability between the different treatment groups.

Mean squares are used as a simplified representation of the data's spread. In short, they provide an interpretation of how widespread the data is either within groups or across all groups.
If MSB is significantly larger than MSW, it might indicate that the treatment differences are the main source of variability, hinting at a potential effect.
F-statistic - Testing for Significance
The F-statistic is a crucial part of ANOVA that determines if the observed group differences are significant. To calculate it, divide the MSB by the MSW. This ratio gives us a number, the F-statistic, which can be interpreted as follows:
  • If the F-statistic is large, it suggests that the variability observed between groups (due to treatments) is much greater than the variability within groups (noise or random error).
  • Conversely, a small F-statistic means there is no significant variance added by the treatments, indicating that most of the observed variation is within groups.

Once calculated, compare the F-statistic to a critical value from the F distribution table, which is determined by dfB and dfW.
If the computed F-statistic exceeds this critical value, we reject the null hypothesis, concluding that there are significant differences between the treatment means.

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Most popular questions from this chapter

The article "The Soundtrack of Recklessness: Musical Preferences and Reckless Behavior Among Adolescents" (Journal of Adolescent Research [1992]: \(313-331\) ) described a study whose purpose was to determine whether adolescents who preferred certain types of music reported higher rates of reckless behaviors, such as speeding, drug use, shoplifting, and unprotected sex. Independently chosen random samples were selected from each of four groups of students with different musical preferences at a large high school: (1) acoustic/pop, (2) mainstream rock, $$\begin{array}{ccccccccc} \text { Type of Box } & & & {\text { Compression Strength (Ib) }} & & & & {\text { Sample Mean }} & {\text { Sample SD }} \\ \hline 1 & 655.5 & 788.3 & 734.3 & 721.4 & 679.1 & 699.4 & 713.00 & 46.55 \\ 2 & 789.2 & 772.5 & 786.9 & 686.1 & 732.1 & 774.8 & 756.93 & 40.34 \\ 3 & 737.1 & 639.0 & 696.3 & 671.7 & 717.2 & 727.1 & 698.07 & 37.20 \\ 4 & 535.1 & 628.7 & 542.4 & 559.0 & 586.9 & 520.0 & 562.02 & 39.87 \\ & & & & & & & \overline{\bar{x}} =682.50 & \end{array}$$ (3) hard rock, and (4) heavy metal. Each student in these samples was asked how many times he or she had engaged in various reckless activities during the last year. The following table lists data and summary quantities on driving over \(80 \mathrm{mph}\) that is consistent with summary quantities given in the article (the sample sizes in the article were much larger, but for the purposes of this exercise, we use \(\left.n_{1}=n_{2}=n_{3}=n_{4}=20\right)\) $$\begin{array}{rrrr} \text { Acoustic/Pop } & \text { Mainstream Rock } & \text { Hard Rock } & \text { Heavy Metal } \\ \hline 2 & 3 & 3 & 4 \\ 3 & 2 & 4 & 3 \\ 4 & 1 & 3 & 4 \\ 1 & 2 & 1 & 3 \\ 3 & 3 & 2 & 3 \\ 3 & 4 & 1 & 3 \\ 3 & 3 & 4 & 3 \\ 3 & 2 & 2 & 3 \\ 2 & 4 & 2 & 2 \\ 2 & 4 & 2 & 4 \\ 1 & 4 & 3 & 4 \\ 3 & 4 & 3 & 5 \\ 2 & 2 & 4 & 4 \\ 2 & 3 & 3 & 5 \\ 2 & 2 & 3 & 3 \\ 3 & 2 & 2 & 4 \\ 2 & 2 & 3 & 5 \\ 2 & 3 & 4 & 4 \\ 3 & 1 & 2 & 2 \\ 4 & 3 & 4 & 3 \\ 20 & 20 & 20 & 20 \\ 2.50 & 2.70 & 2.75 & 3.55 \\ .827 & .979 & .967 & .887 \\ 6830 & 0584 & 0351 & 7868 \end{array}$$ Also, \(N=80\), grand total \(=230.0\), and \(\overline{\bar{x}}=230.0 / 80=\) 2.875. Carry out an \(F\) test to determine if these data provide convincing evidence that the true mean number of times driving over 80 mph varies with musical preference.

The article "Growth Response in Radish to Sequential and Simultaneous Exposures of \(\mathrm{NO}_{2}\) and \(\mathrm{SO}_{2} "(\) Environmental Pollution \([1984]: 303-325\) ) compared a control group (no exposure), a sequential exposure group (plants exposed to one pollutant followed by exposure to the second four weeks later), and a simultaneous-exposure group (plants exposed to both pollutants at the same time). The article states, "Sequential exposure to the two pollutants had no effect on growth compared to the control. Simultaneous exposure to the gases significantly reduced plant growth." Let \(\bar{x}_{1}, \bar{x}_{2}\), and \(\bar{x}_{3}\) represent the sample means for the control, sequential, and simultaneous groups, respectively. Suppose that \(\bar{x}_{1}>\bar{x}_{2}>\bar{x}_{3}\). Use the given information to construct a table where the sample means are listed in increasing order, with those that are judged not to be significantly different underscored. \(15.30\) The nutritional quality of shrubs commonly used for feed by rabbits was the focus of a study summarized in the article "Estimation of Browse by Size Classes for Snowshoe Hare" (Journal of Wildlife Management [1980]: \(34-40\) ). The energy content \((\mathrm{cal} / \mathrm{g})\) of three sizes \((4 \mathrm{~mm}\) or less, \(5-7 \mathrm{~mm}\), and \(8-10 \mathrm{~mm}\) ) of serviceberries was studied. Let \(\mu_{1}, \mu_{2}\), and \(\mu_{3}\) denote the true energy content for the three size classes. Suppose that \(95 \%\) simultaneous confidence intervals for \(\mu_{1}-\mu_{2}, \mu_{1}-\mu_{3}\), and \(\mu_{2}-\mu_{3}\) are \((-10,290),(150,450)\), and \((10,310)\), respectively. How would you interpret these intervals?

An investigation carried out to study purchasers of luxury automobiles reported data on a number of different attributes that might affect purchase decisions, including comfort, safety, styling, durability, and reliability ("Measuring Values Can Sharpen Segmentation in the Luxury Car Market," Journal of Advertising Research \([1995]: 9-\) 22). Here is summary information on the level of importance of speed, rated on a seven-point scale: $$ \begin{array}{lccc} \text { Type of Car } & \text { American } & \text { German } & \text { Japanese } \\ \hline \text { Sample size } & 58 & 38 & 59 \\ \text { Sample mean rating } & 3.05 & 2.87 & 2.67 \end{array} $$ In addition, \(\mathrm{SSE}=459.04\). Carry out a hypothesis test to determine if there is sufficient evidence to conclude that the mean importance rating of speed is not the same for owners of these three types of cars.

Some investigators think that the concentration \((\mathrm{mg} / \mathrm{mL})\) of a particular antigen in supernatant fluids could be related to onset of meningitis in infants. The accompanying data are typical of that given in plots in the article "Type-Specific Capsular Antigen Is Associated with Virulence in Late-Onset Group B Streptococcal Type III Disease" (Infection and Immunity [1984]: 124-129). Construct an ANOVA table, and use it to test the null hypothesis of no difference in mean antigen concentrations for the groups. $$\begin{array}{lllllll} \text { Asymptomatic infants } & 1.56 & 1.06 & 0.87 & 1.39 & 0.71 & 0.87 & \\ \text { Infants with late onset sepsis } & 1.51 & 1.78 & 1.45 & 1.13 & 1.87 & 1.89 & 1.071 .72 \\ \text { Infants with late onset meningitis } & 1.21 & 1.34 & 1.95 & 2.27 & 0.88 & 1.67 & 2.57 \end{array}$$

The article "Heavy Drinking and Problems Among Wine Drinkers" (Journal of Studies on Alcohol [1999]: 467-471) analyzed drinking problems among Canadians. For each of several different groups of drinkers, the mean and standard deviation of "highest number of drinks consumed" were calculated: \(\bar{x}\) $$\begin{array}{lccc} & \overline{\boldsymbol{x}} & \boldsymbol{s} & {n} \\ \hline \text { Beer only } & 7.52 & 6.41 & 1256 \\ \text { Wine only } & 2.69 & 2.66 & 1107 \\ \text { Spirits only } & 5.51 & 6.44 & 759 \\ \text { Beer and wine } & 5.39 & 4.07 & 1334 \\ \text { Beer and spirits } & 9.16 & 7.38 & 1039 \\ \text { Wine and spirits } & 4.03 & 3.03 & 1057 \\ \text { Beer, wine, and spirits } & 6.75 & 5.49 & 2151 \end{array}$$ Assume that each of the seven samples studied can be viewed as a random sample for the respective group. Is there sufficient evidence to conclude that the mean value of highest number of drinks consumed is not the same for all seven groups?
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