Exam 23: More About Analysis of Variance: Follow-Up Tests and Two-Way Anova

A surgeon is interested in studying the range of motion of patients who have hip replacement surgery. He plans to have the patients evaluate their opinion of their range of motion; this will be a subjective score on a five-point scale. The low score, 1, will correspond to zero mobility, and the high score, 5, will correspond to full mobility. An objective measurement of the patient's range of motion will be taken by a physician and will be used as the response. The surgeon believes that the biological sex (male or female) of the patient will also be highly important. It was impossible to force a balanced design, although an equal number of males and females were included and the design was crossed. A total of 60 observations were taken. A partial ANOVA table is provided here: Source DF SS MS F Score 172 Sex 19 Interaction 48 Error 519 C. Total Based on the value of the F statistic for the interaction, and using a 0.05 significance level, what should we conclude about the p-value?

Tail-feather length in birds is sometimes a sexually dimorphic trait. That is, the trait differs substantially for males and for females. Researchers measured tail-feather length (the R1 central tail feather, in mm) in male and female long-tailed finches either raised in an aviary or caught in the wild. This observational study does not have a balanced design, particularly because finches caught in the wild were more difficult to obtain. A total of 52 finches were studied. A partial ANOVA table is provided below, along with an interaction plot displaying the group means. ​ Factor Type Levels Values Malefemale fixed 2 f, m Origin fixed 2 aviary, wild Analysis of Variance for feather length Source DF SS MS F P malefemale 1 2015.5 2015.5 origin 1 251.7 251.7 interaction 1 15.6 15.6 Error 48 4076.9 84.9 S = 9.21600 R-Sq = 47.32% R-Sq(adj) = 44.03% What is the F statistic for the interaction effect?

Tail-feather length in birds is sometimes a sexually dimorphic trait. That is, the trait differs substantially for males and for females. Researchers measured tail-feather length (the R1 central tail feather, in mm) in male and female long-tailed finches either raised in an aviary or caught in the wild. This observational study does not have a balanced design, particularly because finches caught in the wild were more difficult to obtain. A total of 52 finches were studied. A partial ANOVA table is provided below, along with an interaction plot displaying the group means. ​ Factor Type Levels Values Malefemale fixed 2 f, m Origin fixed 2 aviary, wild Analysis of Variance for feather length Source DF SS MS F P malefemale 1 2015.5 2015.5 origin 1 251.7 251.7 interaction 1 15.6 15.6 Error 48 4076.9 84.9 S = 9.21600 R-Sq = 47.32% R-Sq(adj) = 44.03% What is the F statistic for the main effect due to the birds' origin?

Does the environment affect bird weight? Researchers measured the weight (in grams) of male and female adult finches that were either raised in an aviary or caught in the wild. This observational study does not have a balanced design, particularly because finches caught in the wild were more difficult to obtain. A total of 52 finches were studied. A partial ANOVA table is provided below, along with an interaction plot displaying the group means. ​ Factor Type Levels Values malefemale fixed 2 f,m origin fixed 2 aviary, wild Analysis of Variance for bird weight Source DF SS MS F P Malefemale 1 10.844 10.844 origin 1 19.607 19.607 interaction 1 0.340 0.340 Error 48 125.140 2.607 S = 1.61465 R-Sq = 23.88% R-Sq(adj) = 19.12% The first step in analysis is to analyze the interaction effect. Using a significance level of 0.05, what do the results suggest about the interaction effect?

Researchers are interested in the effects of wind direction and season on ozone levels in North America. The ozone levels were recorded at locations, along with the direction of prevailing winds (E, N, S, W) and the season (winter, spring, summer, fall). A balanced design was carried out such that each combination was observed five times. A partial ANOVA table is provided here: What is the mean square for the season main effect? Source DF SS MS F Direction 129 Season 318 Interaction 423 Error 960 C. Total 1830

A study examined the effect of life experience on predator recognition in New Zealand robins. Researchers selected a random sample of continental robins and an independent random sample of island robins. Rats are a natural predator for the continental robins but not for these island robins (due to the recent eradication of rats on the island). The robins were offered five worms either next to a wooden box or next to a fake rat. The time (in minutes) it took each robin to eat all five worms was recorded. The mean eating times are shown for all combinations of location (continental or island) and stimulus (wooden box or fake rat). Wooden box Fake rat Continental 0.75 0.45 Island 0.70 1.60 The P-values for a two-way ANOVA test with an interaction term are as follows: Location: 0.92 Stimulus: 0.011 Location * Stimulus: 0.022 The first step in analysis is to consider the interaction effect. Using a significance level of 0.05, what do the results suggest about the interaction effect?

Tail-feather length in birds is sometimes a sexually dimorphic trait. That is, the trait differs substantially for males and for females. Researchers measured tail-feather length (the R1 central tail feather, in mm) in male and female long-tailed finches either raised in an aviary or caught in the wild. This observational study does not have a balanced design, particularly because finches caught in the wild were more difficult to obtain. A total of 52 finches were studied. A partial ANOVA table is provided below, along with an interaction plot displaying the group means. ​ Factor Type Levels Values Malefemale fixed 2 f, m Origin fixed 2 aviary, wild Analysis of Variance for feather length Source DF SS MS F P malefemale 1 2015.5 2015.5 origin 1 251.7 251.7 interaction 1 15.6 15.6 Error 48 4076.9 84.9 S = 9.21600 R-Sq = 47.32% R-Sq(adj) = 44.03% (Note: If you are using Table F rather than technology, select the nearest degrees of freedom in the table, not necessarily the most conservative value.) What is the P-value for the main effect due to the birds' sex?

Does the environment affect bird weight? Researchers measured the weight (in grams) of male and female adult finches that were either raised in an aviary or caught in the wild. This observational study does not have a balanced design, particularly because finches caught in the wild were more difficult to obtain. A total of 52 finches were studied. A partial ANOVA table is provided below, along with an interaction plot displaying the group means. ​ Factor Type Levels Values malefemale fixed 2 f,m origin fixed 2 aviary, wild Analysis of Variance for bird weight Source DF SS MS F P Malefemale 1 10.844 10.844 origin 1 19.607 19.607 interaction 1 0.340 0.340 Error 48 125.140 2.607 S = 1.61465 R-Sq = 23.88% R-Sq(adj) = 19.12% What is the F statistic for the main effect due to the birds' sex?

Tail-feather length in birds is sometimes a sexually dimorphic trait. That is, the trait differs substantially for males and for females. Researchers measured tail-feather length (the R1 central tail feather, in mm) in male and female long-tailed finches either raised in an aviary or caught in the wild. This observational study does not have a balanced design, particularly because finches caught in the wild were more difficult to obtain. A total of 52 finches were studied. A partial ANOVA table is provided below, along with an interaction plot displaying the group means. ​ Factor Type Levels Values Malefemale fixed 2 f, m Origin fixed 2 aviary, wild Analysis of Variance for feather length Source DF SS MS F P malefemale 1 2015.5 2015.5 origin 1 251.7 251.7 interaction 1 15.6 15.6 Error 48 4076.9 84.9 S = 9.21600 R-Sq = 47.32% R-Sq(adj) = 44.03% (Note: If you are using Table F rather than technology, select the nearest degrees of freedom in the table, not necessarily the most conservative value.) What is the P-value for the main effect due to the birds' origin?

Tail-feather length in birds is sometimes a sexually dimorphic trait. That is, the trait differs substantially for males and for females. Researchers measured tail-feather length (the R1 central tail feather, in mm) in male and female long-tailed finches either raised in an aviary or caught in the wild. This observational study does not have a balanced design, particularly because finches caught in the wild were more difficult to obtain. A total of 52 finches were studied. A partial ANOVA table is provided below, along with an interaction plot displaying the group means. ​ Factor Type Levels Values Malefemale fixed 2 f, m Origin fixed 2 aviary, wild Analysis of Variance for feather length Source DF SS MS F P malefemale 1 2015.5 2015.5 origin 1 251.7 251.7 interaction 1 15.6 15.6 Error 48 4076.9 84.9 S = 9.21600 R-Sq = 47.32% R-Sq(adj) = 44.03% The first step in analysis is to evaluate the interaction effect. Using a significance level of 0.05, what do the results suggest about interaction effect?

A surgeon is interested in studying the range of motion of patients who have hip replacement surgery. He plans to have the patients evaluate their opinion of their range of motion; this will be a subjective score on a five-point scale. The low score, 1, will correspond to zero mobility, and the high score, 5, will correspond to full mobility. An objective measurement of the patient's range of motion will be taken by a physician and will be used as the response. The surgeon believes that the biological sex (male or female) of the patient will also be highly important. It was impossible to force a balanced design, although an equal number of males and females were included and the design was crossed. A total of 60 observations were taken. A partial ANOVA table is provided here: Source DF SS MS F Score 172 Sex 19 Interaction 48 Error 519 C. Total What is the correct null hypothesis for the test of an effect due to the patient's subjective score?

Researchers are interested in the effects of wind direction and season on ozone levels in North America. The ozone levels were recorded at locations, along with the direction of prevailing winds (E, N, S, W) and the season (winter, spring, summer, fall). A balanced design was carried out such that each combination was observed five times. A partial ANOVA table is provided here: Upon rejecting the null hypothesis for the test of an effect due to season, what would be the correct interpretation of the true mean ozone level? Source DF SS MS F Direction 129 Season 318 Interaction 423 Error 960 C. Total 1830

A surgeon is interested in studying the range of motion of patients who have hip replacement surgery. He plans to have the patients evaluate their opinion of their range of motion; this will be a subjective score on a five-point scale. The low score, 1, will correspond to zero mobility, and the high score, 5, will correspond to full mobility. An objective measurement of the patient's range of motion will be taken by a physician and will be used as the response. The surgeon believes that the biological sex (male or female) of the patient will also be highly important. It was impossible to force a balanced design, although an equal number of males and females were included and the design was crossed. A total of 60 observations were taken. A partial ANOVA table is provided here: Source DF SS MS F Score 172 Sex 19 Interaction 48 Error 519 C. Total The physician has determined that an interaction between sex and the subject score is not significant. What would the next step be?

A study examined the effect of life experience on predator recognition in New Zealand robins. Researchers selected a random sample of continental robins and an independent random sample of island robins. Rats are a natural predator for the continental robins but not for these island robins (due to the recent eradication of rats on the island). The robins were offered five worms either next to a wooden box or next to a fake rat. The time (in minutes) it took each robin to eat all five worms was recorded. The mean eating times are shown for all combinations of location (continental or island) and stimulus (wooden box or fake rat). Wooden box Fake rat Continental 0.75 0.45 Island 0.70 1.60 The P-values for a two-way ANOVA test with an interaction term are as follows: Location: 0.92 Stimulus: 0.011 Location * Stimulus: 0.022 The researchers determined there was a significant interaction effect. Now, they will consider the main effects. Using a significance level of 0.05, what do the results suggest about the main effects of location and stimulus?

A study randomly assigned adult subjects to three exercise treatments: (1) a single long exercise period five days per week; (2) several ten-minute exercise periods five days per week; and (3) several ten-minute periods five days per week using a home treadmill. The study report contains the following summary statistics about weight loss (in kilograms) after six months of treatment: Treatment Mean Std. dev. (1) Long periods 10.2 4.2 37 (2) Multiple short periods 9.3 4.5 36 (3) Multiple short periods with treadmill 10.2 5.2 42 The researchers want to contrast the long-period exercise regimen (1) with both short-period regimens (2 and 3). What is the appropriate population contrast?

Does the environment affect bird weight? Researchers measured the weight (in grams) of male and female adult finches that were either raised in an aviary or caught in the wild. This observational study does not have a balanced design, particularly because finches caught in the wild were more difficult to obtain. A total of 52 finches were studied. A partial ANOVA table is provided below, along with an interaction plot displaying the group means. ​ Factor Type Levels Values malefemale fixed 2 f,m origin fixed 2 aviary, wild Analysis of Variance for bird weight Source DF SS MS F P Malefemale 1 10.844 10.844 origin 1 19.607 19.607 interaction 1 0.340 0.340 Error 48 125.140 2.607 S = 1.61465 R-Sq = 23.88% R-Sq(adj) = 19.12% The researcher has determined there is no significant interaction effect, so we may proceed to analyze the main effects of sex and origin. Using a significance level of 0.05, what do the results suggest about the main effects of sex and origin?

Researchers are interested in the effects of wind direction and season on ozone levels in North America. The ozone levels were recorded at locations, along with the direction of prevailing winds (E, N, S, W) and the season (winter, spring, summer, fall). A balanced design was carried out such that each combination was observed five times. A partial ANOVA table is provided here: What is the mean square for the interaction? Source DF SS MS F Direction 129 Season 318 Interaction 423 Error 960 C. Total 1830

Researchers are interested in the effects of wind direction and season on ozone levels in North America. The ozone levels were recorded at locations, along with the direction of prevailing winds (E, N, S, W) and the season (winter, spring, summer, fall). A balanced design was carried out such that each combination was observed five times. A partial ANOVA table is provided here: Based on the F statistic for the interaction, and using a 0.05 significance level, what should we conclude about the p-value for the interaction? Source DF SS MS F Direction 129 Season 318 Interaction 423 Error 960 C. Total 1830

How much corn should be planted per acre for a farmer to get the highest yield? Too few plants will give a low yield, while too many plants will compete with each other for moisture and nutrients, resulting in a lower yield. Four levels of planting density are to be studied: 12,000, 16,000, 20,000, and 24,000 plants per acre. The experimenters had 12 acres available for the study, and three acres were assigned at random to each of the planting densities. A summary of the data follows: ​ ​ Plants (per acre) 12,000 5 2.0 16,000 7 2.8 20,000 12 3.0 24,000 4 1.6 Assume the data are four independent SRSs, one from each of the four populations of planting densities, and that the distribution of the yields is Normal. ​ What is the pooled standard deviation of yield?

How much corn should be planted per acre for a farmer to get the highest yield? Too few plants will give a low yield, while too many plants will compete with each other for moisture and nutrients, resulting in a lower yield. Four levels of planting density are to be studied: 12,000, 16,000, 20,000, and 24,000 plants per acre. The experimenters had 12 acres available for the study, and three acres were assigned at random to each of the planting densities. A summary of the data follows: ​ ​ Plants (per acre) 12,000 5 2.0 16,000 7 2.8 20,000 12 3.0 24,000 4 1.6 Assume the data are four independent SRSs, one from each of the four populations of planting densities, and that the distribution of the yields is Normal. ​ What would be an individual 95% Tukey confidence interval for a difference in the 12,000 and 24,000 planting densities?