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Introduction to the Analysis of Variance

Introduction to the Analysis of Variance

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Introduction to the Analysis of Variance

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  1. Introduction to the Analysis of Variance Basic Concepts, Section 12.1 - 12.2 One-Way ANOVA, Section 12.3

  2. ANOVA Overview • Test for a difference among several means from independently drawn samples • The extension of the two sample t-test for means to three or more samples requires the analysis of variance • Consider the negative income tax experiment in New Jersey • Tested whether was a difference in hours of work between the control and the treatment group • In this experiment income was supplemented by different amounts • The benefit guarantee level ranged from 50 to 125% of the poverty level • Consider then three groups of income • The control group, the first treatment group that received 50% of the poverty level and a second treatment group that received 75% of the poverty level • The null hypothesis is that the mean annual hours over three years is the same for each group • H0: 1 = 2 = 3 • H1: at least one of the population means differs from the others PP 7

  3. ANOVA Overview • Could compare the three population means by evaluating all possible pairs of sample means using the two sample t-test • Compare • Group 1 to group 2 • Group 1 to group 3 • Group 2 to group 3 • For a total of three groups the number of tests required is (3 pick 2) • Evaluated as 3!/(2!1!) • If number of groups = 10 there would be 45 different pair-wise t-tests (10 pick 2) PP 7

  4. ANOVA Overview • Pair-wise t-tests are likely to lead to an incorrect conclusion • Suppose that the three population means are in fact equal and that we conduct all three pair-wise tests • Assume that the tests are independent and set the significance level at 0.05 for each one • By the multiplication rule, the probability of failing to reject a null hypothesis of no difference in all three instances would be • P(fail to reject in all three tests) = (1 - 0.05)3 = (0.95)3 = 0.857 (probability of “accepting” all three) • Consequently, the probability of rejecting the null hypothesis in at least one of the tests is • P(reject in at least one test) = 1 - 0.857 = 0.143 • Since we know that the null hypothesis is true in each case, 0.143 is the probability of committing a type I error PP 7

  5. ANOVA Overview • Need a testing procedure in which the overall probability of committing a Type I error is equal to some predetermined level of alpha • One-way analysis of variance is such a technique • An experiment is a study designed for the purpose of examining the effect that one variable (the independent variable) has on the value of another variable (the dependent variable) PP 7

  6. ANOVA Overview • Negative income tax experiment was a designed experiment • Families were assigned to different treatment groups and given money (or not given money) by the Labor Dept • Intervention by researcher • Hours of work were observed for the next three years • Economists often work with observational studies rather than actual experiments • For example, we might study families from the Current Population Survey or the Census • Observe level of income and hours of work for each family and try to relate the two variables PP 7

  7. ANOVA Overview • In NIT example, hours worked is the dependent variable • What influences the hours of work? • There are three groups of families, distinguished by the amount of income they received from the government • Think of the income received as the independent variable • Income received will influence hours of work • The independent variable is also called the factor or treatment effect • Here we have an experiment in which we try to determine if various levels of a given factor (income) might have different effects on hours of work PP 7

  8. Variation between and within Groups • Looking at the data • There are three different levels of the factor income • The values of the hours worked for the different families are grouped by the factor level • We observe the group means Factor: Income Supplement Levelj groups, j=1, 2, …t i rows, i=1, 2, …n 123 Measurements: x11 x12 x13 Hours Worked x21 x22 x23 for different families .. .. .. xn1 xn2 xnt Group Mean PP 7

  9. Two Sources of Variation • Variation between groups reflects the effect of the factor levels, of the treatment • Variation between groups is seen by looking at the three group means • If there are large differences in the group means • Suggest that the differences in income supplements has an effect on average hours worked • Variation within groups represents random error from sampling • Values within a sample will vary chance • ANOVA uses these two kinds of variation to test for whether the factor has an effect on the dependent variable PP 7

  10. The Model and Assumptions • One-way analysis of variance • Examines populations that are classified by one characteristic • In our example, the characteristic is the amount of income supplement the family receives • There are three levels of that factor, or three groups • If we had only two samples instead of t samples, one-way ANOVA is equivalent to the two sample equal variance t-test for independent samples PP 7

  11. Assumptions • The samples have been independently selected • The population variances are equal • Not usually tested • The dependent variable follows a normal distribution in the populations PP 7

  12. Online Homework - Chapter 12 Intro to ANOVA • CengageNOW ninth assignment: Chapter 12 Intro to ANOVA PP 7

  13. Procedure • Remember each population represents a level of a factor • The hypotheses are • H0: 1 = 2 = …. = t • H1: Not all the means are equal • The null hypothesis would be • Supported if we observed small differences from one sample mean to the next • Rejected if at least some of the differences in sample means were large PP 7

  14. Procedure • We need a precise measure of the discrepancies among the sample means • A possible choice is the variance of the sample means • The basic idea of ANOVA is to express a measure of the total variation in a data set as a sum of two components • Variation within groups and variation between groups • If the variation within groups is small relative to the variation between the group means • Suggests that the population means are in fact different PP 7

  15. Problem – Are There Any Differences in Detergents? • Consumer Report is testing the cleansing action of three leading detergents • Cleansing action is the dependent variable • The different detergents represent the treatment • There are three levels of the factor because there are three detergents PP 7

  16. Problem – Are There Any Differences in Detergents? • There are 15 swatches of dirty cloth • We select at random 5 swatches to be washed by each of the detergents • After the swatches are cleaned, rate each on the basis of 0 to 100 • Let the level of significance be 0.01 PP 7

  17. Problem – Are There Any Differences in Detergents? What is the value of x23? What is x51? = X23 = x51 PP 7

  18. Problem – Are There Any Differences in Detergents? • Consider all 15 observations as one data set for the moment • Calculate the total variation in the pooled data set • Then break the total variation into two component • Variation within groups • Variation between groups PP 7

  19. Total Variation • Total variation • Grand mean • Where xij is the ith observation in the jth sample • j = 1, 2,….t samples or groups or levels of the factor • i = 1, 2, … nj observations in a group PP 7

  20. Total Variation • Grand mean is the mean of all the pooled observations • Capital N represents the total number of observations when the data are pooled • Not necessary for each sample (group) to have the same number of observations PP 7

  21. Summation Notation - Grand Mean • When we work with double summation signs, evaluate the inner summation sign first PP 7

  22. Total Sum of Squares • The total sum of squares can be found next, SST PP 7

  23. SST = SSTR + SSE • SST is divided into the variation between groups and the variation within groups (not variance) • SST = SSTR + SSE • SSTR = Variation between groups (Treatment) • SSE = Variation within groups (Error) • SST = SSB + SSW PP 7

  24. SSTR – Treatment Sum of Squares -Variation between Groups • SSTR The dot means that the average is carried out across the index i. We select a particular group, j, and then find the average of all the observations within that group. PP 7

  25. SSE Note that SST = SSTR + SSE 666 = 390 + 276 Can solve for two of the three and find the remaining Sum of Squares (SS) by subtraction SSE - Error Sum of Squares - Variation within Groups PP 7

  26. SST = SSTR + SSE • Examine two different variances • One based on the SSTR • The other based on the SSE • Remember that a variance is computed by dividing the sum of squared deviations by the appropriate degrees of freedom • Do the same here • Create Variances • Also called Mean Squared Deviations PP 7

  27. Mean Square Deviations • Mean Square Deviation for Treatment where t = the number of groups (We use up one degree of freedom in estimating the grand mean.) where N = the total number of observations across all groups (Each group mean is estimated by the sample observations anduses up one degree of freedom.) PP 7

  28. Rationale of the Test • The variance within groups, MSE, measures • Variability of the values around the mean of each group • Random variation of values within groups • The variance between groups, MSTR, measures • Random variation of values within groups • Also measures differences from one group to another • If there is no real difference from group to group, the variance between groups should be close to the variance within groups • MSTR  MSE • Ratio is close to 1 • However, if there is a difference between groups, then • MSTR > MSE PP 7

  29. = Ft-1,N-t ANOVA - Test Statistic • Test Statistic • If the null hypothesis is true and we draw a large number of samples from the populations and calculate the test statistic repeatedly • The sampling distribution of the test statistic follows the F distribution with t - 1 and N - t degrees of freedom • “Most” of the F values will be close to 1 PP 7

  30. reject = Ft – 1,N - t ANOVA - Test Statistic Sampling Distribution of • Even when the null hypothesis is true, arithmetically, the SSTR > SSE • So the test takes place in the upper tail of the distribution • Place all of the level of significance in the upper tail PP 7

  31. = Ft – 1,N - t ANOVA – Test Statistic Sampling Distribution of • Find critical value F⍺ • The decision rule is • If test statistic reject the H0 reject PP 7

  32. 0.01 Problem - ANOVA Sampling Distribution of • Calculate the MSTR • Calculate the MSE • Calculate the F test Do not reject reject F2,12 PP 7

  33. F Table ⍺ = 0.01

  34. 0.01 6.93 8.48 F2,12 Problem - ANOVA • Find critical value at  = 0.01 • Reject H0, some of the means differ significantly • Some of the detergents clean better than others reject PP 7

  35. ANOVA Table PP 7

  36. Completed ANOVA Table PP 7

  37. ANOVA p - value • Computer output provides the probability of observing an F test statistic as large as 8.48 if the H0 is true • This p-value is 0.0051 • To find the p-value, in a cell within a Microsoft Excel spreadsheet, type • =FDIST(Test value, t-1, N-t) • =FDIST(8.48,2,12) = .0051 • Setting our level of significance at 0.01, .0051 < 0.01 • Reject the null hypothesis PP 7

  38. Multiple Comparison Procedures • What happens if we reject the null hypothesis? • Conclude that the population means are not all equal • Do not know whether all of the means are different from one another or if only some of them are different • Want to conduct additional tests to find out where the differences lie • Number of multiple comparison tests available, each with advantages and disadvantages • Simple approach is to perform a series of two sample t-tests • This increases the probability of committing a Type I error • Avoid this problem by reducing the individual  levels to ensure that the overall level of significance is kept at a predetermined level PP 7

  39. ANOVA Assumption - Homogeneity of Variances • Bartlett’s Test for Homogeneity of Variances • Most common method used to test whether the population variances are equal • Test is powerful • Can discern that the null hypothesis is false • Badly affected by non-normal populations • ANOVA is robust • Robust means that the validity of a test is not seriously affected by moderate deviations from the underlying assumptions • Anova operates well even with considerable heterogeneity of variances, as long as nj are equal or nearly equal • ANOVA is also robust with respect to the assumption of the underlying populations’ normality, especially as n increases PP 7

  40. Online Homework - Chapter 12 ANOVA • CengageNOW tenth assignment: Chapter 12 ANOVA • CengageNOW eleventh assignment: Chapter 12: Overview of ANOVA PP 7

  41. Multiple Comparison Technique: Bonferroni Correction • The significance level for each of the individual comparisons depends on the number of pair-wise tests being conducted • In our problem, we set  = 0.01 and we have (3 pick 2) = 3 pair-wise comparisons • To set the overall probability of committing a Type I error at 0.01 we should usefor the significance level for an individual comparison PP 7

  42. Bonferroni Correction • Instead of pooling the data from only two samples to estimate the common variance, pool all t samples • Degrees of freedom are N – t • The test statistic is PP 7

  43. Bonferroni Correction • The sample variances are • The pooled variance is S1 = 3.937 S2 = 6.325 S3 = 3.674 PP 7

  44. Bonferroni Correction: Group 1&2, Group 1&3, Group 2&3 • Perform three t –tests p-value = .0118, do not reject at  = .003 p-value = .171, do not reject at  = .003 p-value = .0019, reject at  = 0.003. There is a significant difference between detergent 2 and 3. PP 7

  45. P-values from Excel • Using Excel’s statistical function • =TDIST(x,df,tails) • =TDIST(2.967,12,2) • =TDIST(-.989,12,2) • =TDIST(-3.956,12,2) PP 7