Two-Way Analysis of Variance: An Example with Granola Bars and Blood Pressure Data - Prof., Study notes of Data Analysis & Statistical Methods

Download Two-Way Analysis of Variance: An Example with Granola Bars and Blood Pressure Data - Prof. and more Study notes Data Analysis & Statistical Methods in PDF only on Docsity! Chapter 13: Two-Way Analysis of Variance Chapter 7: Two-sample comparison of means t tests (1 categorical and 1 quantitative variables) Example: Are the mean taste ratings of chewy granola bars the same as those for crunchy granola bars if you conduct a taste test (scale of 1-10)? Chapter 12: F tests compare the means of several populations (1 categorical and 1 quantitative variables) Example: Are the mean taste ratings of Quaker, Kellogg’s, and Nature Valley granola bars the same if you conduct a taste test (scale of 1-10)? Chapter 13: F tests compare the means of populations that are classified in 2 ways (2 categorical and 1 quantitative variables) Example: Do brand, texture (chewy vs. crunchy), and/or their interaction make a difference to the mean taste ratings (scale of 1-10) for granola bars? What’s similar for Two-Way ANOVA? Just as in One-way ANOVA we still: • assume the data are approximately normal • the groups have the same standard deviation (even if the means may be different) • pool to estimate the standard deviation • use F statistics for significance tests. What’s different for Two-Way ANOVA? We can look at each categorical variable separately, and we can look at their interaction. (With one-way ANOVA it was impossible to look at interaction.) 1 Example (from 4th edition of M&M): Each of the following situations is a 2-way study design. For each case, identify the response variable and both factors, and state the number of levels for each factor (I and J) and the total number of observations (N). a) A study of smoking classifies subjects as nonsmokers, moderate smokers, or heavy smokers. Samples of 80 men and 80 women are drawn from each group. Each person reports the number of hours of sleep he or she gets on a typical night. b) The strength of concrete depends upon the formula used to prepare it. An experiment compares 6 different mixtures. Nine specimens of concrete are poured from each mixture. Three of these specimens are subjected to 0 cycles of freezing and thawing, 3 are subjected to 100 cycles, and 3 specimens are subjected to 500 cycles. The strength of each specimen is then measured. c) Four methods for teaching sign language are to be compared. Sixteen students in special education and sixteen students majoring in other areas are the subjects for the study. Within each group they are randomly assigned to the methods. Scores on a final exam are compared. Example (from 4th edition of M&M): In the course of a clinical trial of measures to prevent coronary heart disease, blood pressure measurements were taken on 12,866 men. Individuals were classified by age group and race. The means for systolic blood pressure are given in the following table: 35-39 40-44 45-49 50-54 55-59 White 131 132.3 135.2 139.4 142 Non-White 132.3 134.2 137.2 141.3 144.1 Note that we are not given raw data on these 12,866 men. The table above is the mean for each race/age combination. This means we can’t use ANOVA. We’ll just use graphing and marginal means to describe the situation. 2 Example (Exercise 13.20): One way to repair serious wounds is to insert some material as a scaffold for the body’s repair cells to use as a template for new tissue. Scaffolds made from extracellular material (ECM) are particularly promising for this purpose. Because they are made from biological material, they serve as an effective scaffold and are then reabsorbed. One study compared 6 types of scaffold material. Three of these were ECMs and the other three were made of inert materials. There were 3 mice used per scaffold type. The response measure was the % of glucose phosphated isomerase (Gpi) cells in the region of the wound. A large value is good, indicating that there are many bone marrow cells sent by the body to repair the tissue. Here are the data for 2 weeks, 4 weeks, and 8 weeks after the repair: Gpi % Material 2 weeks 4 weeks 8 weeks 70 55 60 75 70 65 ECM1 65 70 65 60 60 60 65 65 70 ECM2 70 65 60 80 75 70 60 70 80 ECM3 75 75 70 50 20 15 45 25 25 MAT1 50 25 25 5 5 10 10 10 5 MAT2 15 5 5 30 10 5 25 15 15 MAT3 25 10 10 Using SPSS: Analyze General Linear Model Univariate. Move “Gpi” into “Dependent Variable” box. Move “Time” and “Material” into “Fixed Factor(s)” box. To get a means plot, click “Plots” box, move “Material” into the “Horizontal Axis” box. Move “Time” into the “Separate Lines” box. Click “Add” and “Continue.” To get summary statistics, click “Options” box. Move “Material” and “Time” into the “Display Means” box. Click “Descriptive Statistics” box, and click “Continue.” Click “OK.” 5 a) Make a table giving the sample size, mean, and standard deviation for each of the material-by-time combinations. Is it reasonable to pool the variances? Descriptive Statistics Dependent Variable: GPI 70.00 5.000 3 65.00 8.660 3 63.33 2.887 3 66.11 6.009 9 66.67 7.638 3 63.33 2.887 3 63.33 5.774 3 64.44 5.270 9 71.67 10.408 3 73.33 2.887 3 73.33 5.774 3 72.78 6.180 9 48.33 2.887 3 23.33 2.887 3 21.67 5.774 3 31.11 13.411 9 10.00 5.000 3 6.67 2.887 3 6.67 2.887 3 7.78 3.632 9 26.67 2.887 3 11.67 2.887 3 10.00 5.000 3 16.11 8.580 9 48.89 24.648 18 40.56 28.330 18 39.72 28.619 18 43.06 27.064 54 Time 2 4 8 Total 2 4 8 Total 2 4 8 Total 2 4 8 Total 2 4 8 Total 2 4 8 Total 2 4 8 Total Material ECM1 ECM2 ECM3 MAT1 MAT2 MAT3 Total Mean Std. Deviation N (Because the sample sizes in this experiment are very small, we expect a large amount of variability in the sample standard deviations. Although they vary more than we would prefer, we will proceed with the ANOVA.) 6 b) Make a plot of the means of the combinations. Describe the main features of the plot. c) Make a table of the sample size, and mean for each type of material. Make a plot of the means of the materials. Give a short summary of the % Gpi depends on the type of material. 1. Material Dependent Variable: GPI 66.111 1.742 62.578 69.644 64.444 1.742 60.911 67.978 72.778 1.742 69.245 76.311 31.111 1.742 27.578 34.644 7.778 1.742 4.245 11.311 16.111 1.742 12.578 19.644 Material ECM1 ECM2 ECM3 MAT1 MAT2 MAT3 Mean Std. Error Lower Bound Upper Bound 95% Confidence Interval 7