Stat 112: Lecture 22 Notes • Chapter 9.1: One-way Analysis of Variance. • Chapter 9.3: Two-way Analysis of Variance • Homework 6 is due on Friday.
Errors in Hypothesis Testing When we do one hypothesis test and reject null hypothesis if p-value <0.05, then the probability of making a Type I error when the null hypothesis is true is 0.05. We protect against falsely rejecting a null hypothesis by making probability of Type I error small.
Multiple Comparisons Problem • Compound uncertainty: When doing more than one test, there is an increased chance of a Type I error • If we do multiple hypothesis tests and use the rule of rejecting the null hypothesis in each test if the p-value is <0.05, then if all the null hypotheses are true, the probability of falsely rejecting at least one null hypothesis is >0.05.
Individual vs. Familywise Error Rate • When several tests are considered simultaneously, they constitute a family of tests. • Individual Type I error rate: Probability for a single test that the null hypothesis will be rejected assuming that the null hypothesis is true. • Familywise Type I error rate: Probability for a family of test that at least one null hypothesis will be rejected assuming that all of the null hypotheses are true. • When we consider a family of tests, we want to make the familywise error rate small, say 0.05, to protect against falsely rejecting a null hypothesis.
Bonferroni Method • General method for doing multiple comparisons for any family of k tests. • Denote familywise type I error rate we want by p*, say p*=0.05. • Compute p-values for each individual test -- • Reject null hypothesis for ith test if • Guarantees that familywise type I error rate is at most p*. • Why Bonferroni works: If we do k tests and all null hypotheses are true , then using Bonferroni with p*=0.05, we have probability 0.05/k to make a Type I error for each test and expect to make k*(0.05/k)=0.05 errors in total.
Bonferroni on Milgram’s Data Output obtained from Fit Y by X, Compare Means, Each Pair Student’s t
Tukey’s HSD • Tukey’s HSD is a method that is specifically designed to control the familywise type I error rate (at 0.05) for analysis of variance when we are interested in comparing all pairs of groups. • JMP Instructions: After Fit Y by X, click the red triangle next to the X variable and click LSMeans Tukey HSD.
Assumptions in one-way ANOVA • Assumptions needed for validity of one-way analysis of variance p-values and CIs: • Linearity: automatically satisfied. • Constant variance: Spread within each group is the same. • Normality: Distribution within each group is normally distributed. • Independence: Sample consists of independent observations.
Rule of thumb for checking constant variance • Constant variance: Look at standard deviation of different groups by using Fit Y by X and clicking Means and Std Dev. • Rule of Thumb: Check whether (highest group standard deviation/lowest group standard deviation) is greater than 2. If greater than 2, then constant variance is not reasonable and transformation should be considered.. If less than 2, then constant variance is reasonable. • (Highest group standard deviation/lowest group standard deviation) =(131.874/63.640)=2.07. Thus, constant variance is not reasonable for Milgram’s data.
Transformations to correct for nonconstant variance • If standard deviation is highest for high groups with high means, try transforming Y to log Y or . If standard deviation is highest for groups with low means, try transforming Y to Y2. • SD is particularly low for group with highest mean. Try transforming to Y2. To make the transformation, right click in new column, click New Column and then right click again in the created column and click Formula and enter the appropriate formula for the transformation.
Transformation of Milgram’s data to Squared Voltage Level • Check of constant variance for transformed data: (Highest group standard deviation/lowest group standard deviation) = 1.63. Constant variance assumption is reasonable for voltage squared. • Analysis of variance tests are approximately valid for voltage squared data; reanalyzed data using voltage squared.
Analysis using Voltage Squared Strong evidence that the group mean voltage squared levels are not all the same. Strong evidence that remote has higher mean voltage squared level than proximity and touch-proximity and that voice-feedback has higher mean voltage squared level than touch-proximity, taking into account the multiple comparisons.
Rule of Thumb for Checking Normality in ANOVA • The normality assumption for ANOVA is that the distribution in each group is normal. Can be checked by looking at the boxplot, histogram and normal quantile plot for each group. • If there are more than 30 observations in each group, then the normality assumption is not important; ANOVA p-values and CIs will still be approximately valid even for nonnormal data if there are more than 30 observations in each group. • If there are less than 30 observations per group, then we can check normality by clicking Analyze, Distribution and then putting the Y variable in the Y, Columns box and the categorical variable denoting the group in the By box. We can then create normal quantile plots for each group and check that for each group, the points in the normal quantile plot are in the confidence bands. If there is nonnormality, we can try to use a transformation such as log Y and see if the transformed data is approximately normally distributed in each group.
One way Analysis of Variance: Steps in Analysis • Check assumptions (constant variance, normality, independence). If constant variance is violated, try transformations. • Use the effect test (commonly called the F-test) to test whether all group means are the same. • If it is found that at least two group means differ from the effect test, use Tukey’s HSD procedure to investigate which groups are different, taking into account the fact multiple comparisons are being done.
Analysis of Variance Terminology • The criterion (criteria) by which we classify the groups in analysis of variance is called a factor. In one-way analysis of variance, we have one factor. • The possible values of the factor are levels. • Milgram’s study: Factor is experimental condition with levels remote, voice-feedback, proximity and touch-proximity. • Two-way analysis of variance: Groups are classified by two factors.
Two-way Analysis of Variance Examples • Milgram’s study: In thinking about the Obedience to Authority study, many people have thought that women would react differently than men. Two-way analysis of variance setup in which the two factors are experimental condition (levels remote, voice-feedback, proximity, touch-proximity) and sex (levels male, female). • Package Design Experiment: Several new types of cereal packages were designed. Two colors and two styles of lettering were considering. Each combination of lettering/color was used to produce a package, and each of these combinations was test marketed in 12 comparable stores and sales in the stores were recorded.. Two-way analysis of variance in which two factors are color (levels red, green) and lettering (levels block, script). • Goal of two-way analysis of variance: Find out how the mean response in a group depends on the levels of both factors and find the best combination.
Two-way Analysis of Variance • The mean of the group with the ith level of factor 1 and the jth level of factor 2 is denoted , e.g., in package-design experiment, the four group means are • As with one-way analysis of variance, two-way analysis of variance can be seen as a a special case of multiple regression. For two-way analysis of variance, we have two categorical explanatory variables for the two factors and also include an interaction between the factors.
Estimated Mean for Red Block group = 144.92+9.83-11.17+4.5 = 148.08 Estimated Mean for Red Script group = 144.92+9.83+11.17-4.5= 161.42
The LS Means Plots show how the means of the groups vary as the levels of the factors vary. For the top plot for color, green refers to the mean of the two green groups (green block and green script) and red refers to the mean of the two red groups (red block and red script). Similarly for the second plot for TypeStyle, block refers to the mean of the two block groups (red block and green block). The third plot for TypeStyle*Color shows the mean of all four groups.
Two-way ANOVA in JMP • Use Analyze, Fit Model with a categorical variable for the first factor, a categorical variable for the second factor and an interaction variable that crosses the first factor and the second factor. • The LS Means Plots are produced by going to the output in JMP for each variable that is to the right of the main output, clicking the red triangle next to each variable (for package design, the vairables are Color, TypeStyle, Typestyle*Color) and clicking LS Means Plot.
Interaction in Two-Way ANOVA • Interaction between two factors: The impact of one factor on the response depends on the level of the other factor. • For package design experiment, there would be an interaction between color and typestyle if the impact of color on sales depended on the level of typestyle. • Formally, there is an interaction if • LS Means Plot suggests there is not much interaction. Impact of changing color from red to green on mean sales is about the same when the typestyle is block as when the typestyle is script.
Effect Test for Interaction • A formal test of the null hypothesis that there is no interaction, for all levels i,j,i’,j’ of factors 1 and 2, versus the alternative hypothesis that there is an interaction is given by the Effect Test for the interaction variable (here Typestyle*Color). • p-value for Effect Test = 0.4191. No evidence of an interaction.
Implications of No Interaction • When there is no interaction, the two factors can be looked in isolation, one at a time. • When there is no interaction, best group is determined by finding best level of factor 1 and best level of factor 2 separately. • For package design experiment, suppose there are two separate groups: one with an expertise in lettering and the other with expertise in coloring. If there is no interaction, groups can work independently to decide best letter and color. If there is an interaction, groups need to get together to decide on best combination of letter and color.
Model when There is No Interaction • When there is no evidence of an interaction, we can drop the interaction term from the model for parsimony and more accurate estimates: Mean for red block group = 144.92+9.83-11.17=143.58 Mean for red script group = 144.92+9.83+11.17=165.92
Tests for Main Effects When There is No Interaction • Effect test for color: Tests null hypothesis that group mean does not depend on color versus alternative that group mean is different for at least two levels of color. p-value =0.0804, moderate but not strong evidence that group mean depends on color. • Effect test for TypeStyle: Tests null hypothesis that group mean does not depend on TypeStyle versus alternative that group mean is different for at least two levels of TypeStyle. p-value = 0.0481, evidence that group mean depends on TypeStyle. • These are called tests for “main effects.” These tests only make sense when there is no interaction.
Example with an Interaction • Should the clerical employees of a large insurance company be switched to a four-day week, allowed to use flextime schedules or kept to the usual 9-to-5 workday? • The data set flextime.JMP contains percentage efficiency gains over a four week trial period for employees grouped by two factors: Department (Claims, Data Processing, Investment) and Condition (Flextime, Four-day week, Regular Hours).
Which schedule is best appears to differ by department. Four day is best for investment employees, but worst for data processing employees.
Which Combinations Works Best? • For which pairs of groups is there strong evidence that the groups have different means – is there strong evidence that one combination works best? • We combine the two factors into one factor (Combination) and use Tukey’s HSD, to compare groups pairwise, adjusting for multiple comparisons.
Checking Assumptions • As with one-way ANOVA, two-way ANOVA is a special case of multiple regression and relies on the assumptions: • Linearity: Automatically satisfied • Constant variance: Spread within groups is the same for all groups. • Normality: Distribution within each group is normal. • To check assumptions, combine two factors into one factor (Combination) and check assumptions as in one-way ANOVA.
Checking Assumptions • Check for constant variance: (Largest standard deviation of group/Smallest standard deviation of group)=(44.85/33.51)<2. Constant variance OK. • Check for normality: Look at normal quantile plots for each combination (not shown). For all normal quantile plots, the points fall within the 95% confidence bands. Normality assumption OK.
Two way Analysis of Variance: Steps in Analysis • Check assumptions (constant variance, normality, independence). If constant variance is violated, try transformations. • Use the effect test (commonly called the F-test) to test whether there is an interaction. • If there is no interaction, use the main effect tests to whether each factor has an effect. Compare individual levels of a factor by using t-tests with Bonferroni correction for the number of comparisons being made. • If there is an interaction, use the interaction plot to visualize the interaction. Create combination of the factors and use Tukey’s HSD procedure to investigate which groups are different, taking into account the fact multiple comparisons are being done.