ࡱ>  yl[ qbjbj 4&ΐΐh||,hb %%%%(((aaaaaaa$LdfJa-('|(((a%% "b+++(^%%a+(a++Yl^%p#OTn+7[*a8b0hba[8gn+:8gT^8g^((+(((((aa+(((hb((((8g(((((((((| : One-Way Analysis of Variance (ANOVA) The basic scenario leading to an analysis of variance is one in which the goal is to explain the effect of one or more qualitative (categorical) variables upon a single quantitative variable. In these notes we assume a single qualitative independent variable, or a one-way analysis of variance. Two equivalent models are commonly employed in one-way ANOVA: the cell means model, and the factor effects model. Although these notes primarily cover the cell means model, we will refer to the factor effects model where convenient and you are expected be able to move between model descriptions. Cell Means Model: The following is meant to explain the notation used in the cell means model: Factor: Another name for the single qualitative independent variable. Level: one of the k distinct values of the factor involved in the analysis. Also called a Treatment.  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , is the size of the sample drawn from the ith treatment population of the factor.  EMBED Equation.DSMT4 is the total sample size drawn from all treatments.  EMBED Equation.DSMT4 : The mean value of the quantitative dependent (response) variable under the ith treatment.  EMBED Equation.DSMT4 : The value of the response variable for the jth observation from the ith treatment in the sample.  EMBED Equation.DSMT4 : The difference between  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 , i.e.,  EMBED Equation.DSMT4 . Note: The assumption made in ANOVA is that the errors are independent and identically distributed  EMBED Equation.DSMT4 .  For the jth observation on the ith treatment, the cell means model is given by  EMBED Equation.DSMT4  . Since the treatment mean  EMBED Equation.DSMT4 is a constant, we have  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . Finally, this leads to the conclusion that, under the assumptions made about the distribution of the errors, the  EMBED Equation.DSMT4  are independent  EMBED Equation.DSMT4 . Factor Effects Model: The following is meant to explain the notation used in the factor effects model:  EMBED Equation.DSMT4 : Under the most common interpretation, this is the unweighted average of all treatment means, i.e.,  EMBED Equation.DSMT4 . (In some situations,  EMBED Equation.DSMT4 may be defined as a weighted average of the treatment means.)  EMBED Equation.DSMT4 : The effect of the ith treatment. If  EMBED Equation.DSMT4 is the unweighted mean of all treatment means, then  EMBED Equation.DSMT4 .  For the jth observation on the ith treatment, the factor effects model is given by  EMBED Equation.DSMT4  . Since the treatment mean  EMBED Equation.DSMT4 is a constant, we have  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . Finally, this leads to the conclusion that, under the assumptions made about the distribution of the errors, the  EMBED Equation.DSMT4  are independent  EMBED Equation.DSMT4 . Fitting a One-Way ANOVA Model The Analysis of Variance is a linear model because it can be expressed in the form  EMBED Equation.DSMT4 , where the forms of the design matrix  EMBED Equation.DSMT4 and the coefficient vector  EMBED Equation.DSMT4  depend upon whether the cell means or factor effects models are being fitted. It can be shown that the least squares solution to the problem of fitting an ANOVA model to a sample is given by: Cell Means Model: For the cell means model, under the least squares criterion, we seek to minimize the sum  EMBED Equation.DSMT4 . This can be rewritten  EMBED Equation.DSMT4 . The only way to minimize the overall sum is to minimize each of the component sums. You will show in a homework problem that a sum of the form  EMBED Equation.DSMT4 is minimized by letting  EMBED Equation.DSMT4 . Thus, the least squares estimates of the treatment means are the corresponding sample means,  EMBED Equation.DSMT4 . The following notation is used for the cell means model:  EMBED Equation.DSMT4 : The mean response for the sample drawn from the ith treatment population. So,  EMBED Equation.DSMT4   EMBED Equation.DSMT4 : The mean response for the total sample,  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4 is the total number of observations made on the dependent variable Y.  EMBED Equation.DSMT4 : The residual for the jth observation on the ith treatment,  EMBED Equation.DSMT4 . Factor Effects Model: For the factor effects model, under the least squares criterion, the estimates are the same as those obtained in fitting the cell means model, but the statistics being fitted differ slightly. In particular,  EMBED Equation.DSMT4 : the estimated effect of the ith treatment on the mean of the response variable, given by  EMBED Equation.DSMT4 . Partitioning the Sums of Squares In both the cell means and the factor effects model, we'll differentiate between the following deviations:  EMBED Equation.DSMT4 is the deviation of the jth observation from the ith treatment from the overall mean for the sample. (This corresponds to  EMBED Equation.DSMT4 in regression.)  EMBED Equation.DSMT4 is the deviation of the sample mean for the ith treatment from the overall mean for the sample. (This corresponds to  EMBED Equation.DSMT4  in regression.)  EMBED Equation.DSMT4 is the deviation of the jth observation from the ith treatment from the sample mean for the treatment. (This corresponds to  EMBED Equation.DSMT4  in regression.) The deviations above each lead to a sum of squares corresponding to a familiar sum of squares in regression.  EMBED Equation.DSMT4  is the Total Sum of Squares in one-way ANOVA.  EMBED Equation.DSMT4  is the "Between Treatments" Sum of Squares. (Note: traditionally, this sum of squares goes by other names, such as the treatment sum of squares, for example. However, since there is no generally accepted choice of name, I've decided to use the name it would have in regression.)  EMBED Equation.DSMT4  is the Error Sum of Squares, just as in regression. It's also called the "Within Treatments" sum of squares because it measures error from the treatment means. If we accept for the moment that sums of squares for one-way ANOVA correspond exactly to their counterparts in regression, then it won't come as a surprise that the sums of squares add similarly: SST = SSR + SSE . We can also partition the degrees of freedom as follows:  EMBED Equation.DSMT4  , because we lose one of the n "bits" of information contained in the n observations when we must estimate the overall mean  EMBED Equation.DSMT4 by  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  , because we must estimate k - 1 treatment means  EMBED Equation.DSMT4 after we've estimated the overall mean  EMBED Equation.DSMT4 . (Note: the estimates of  EMBED Equation.DSMT4 and of the k treatment means EMBED Equation.DSMT4 are not independent because  EMBED Equation.DSMT4  can be obtained from the k estimates  EMBED Equation.DSMT4 .)  EMBED Equation.DSMT4  Note: the degrees of freedom partition in the same way as do the corresponding sums of squares:  EMBED Equation.DSMT4 . Although we haven't proved it, this is a result of the geometry of the vector space of observations, EMBED Equation.DSMT4 . The vector of deviations "Between Treatments" and the vector of deviations "Within Treatments" lie in orthogonal subspaces in  EMBED Equation.DSMT4 . Thus we obtain the same Pythagorean right triangle in  EMBED Equation.DSMT4 seen in regression, where the "Between" and "Within" vectors form the legs, and the vector of "Total" deviations forms the hypotenuse. Thus, the sums of squares relation SST = SSR + SSE is just a statement of the Pythagorean Theorem in  EMBED Equation.DSMT4 , where SSR and SSE are the squared lengths of the "Between" and "Within" vectors, respectively. Finally, the degrees of freedom relation  EMBED Equation.DSMT4  is just a statement of the dimensionality of the subspaces where the vectors "live." The table below summarizes the relationships between the sums of squares and their degrees of freedom for both a one-way analysis of variance with k treatments, and a multiple regression with k-1 independent variables. Question: why is the One-Way ANOVA with k levels equivalent to a multiple regression with only k-1 independent variables? One-Factor ANOVA with k levelsMultiple Regression with k - 1 independent variablesSums of SquaresdfSums of SquaresdfSSR "Between"k-1SSRk-1SSE "Within"n-kSSEn-kSST "Total"n-1SSTn-1 Testing for a Differential Treatment Effect on the Response Variable The hypothesis test for treatment effect has the following forms, both of which lead to the same ANOVA table: Cell Means Model:  EMBED Equation.DSMT4 . This is claim that all levels of the factor variable produce the same effect on the mean of the response variable.  EMBED Equation.DSMT4 At least two of the treatment means differ. Factor Effects Model:  EMBED Equation.DSMT4 . This is the same null hypothesis as the cell means model, but considering the differential treatment effects  EMBED Equation.DSMT4 instead of the treatment means  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4 At least two  EMBED Equation.DSMT4  are not zero. Equivalently, at least two of the treatment means differ. All Roads (Really Do) Lead to F: the F - Ratio in One-Way ANOVA Chi-Square Random Variables and the F Distribution If the errors in a one-way analysis of variance model are independent EMBED Equation.DSMT4 random variables, then  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4 is the Chi-square distribution with k - 1 degree of freedom. Similarly,  EMBED Equation.DSMT4 . Note: The degrees of freedom above agree with those resulting from a multiple regression model based on one qualitative variable with k levels. Remember, we would only create k - 1 dummy variables for the k distinct values of the factor. Thus the k - 1 degrees of freedom for SSR reflects the k - 1 "slopes" estimated in the model Y = b0 + b1X1 + b2X2 + & + bk-1Xk-1 + e. Then the F - Ratio,  EMBED Equation.DSMT4 , that appears in the ANOVA table is the ratio of two independent chi-square distributions divided by their respective degrees of freedom. Under the model assumptions, the F - Ratio follows an F distribution with degrees of freedom EMBED Equation.DSMT4 and EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4 and EMBED Equation.DSMT4  are the degrees of freedom of the chi-square variables in the numerator and denominator of the F - Ratio, respectively. The F - Ratio for One-Way Analysis of Variance (the ANOVA Table) The hypothesis for the test in one-way ANOVA takes the form  EMBED Equation.DSMT4 All treatment means equal, versus,  EMBED Equation.DSMT4 Some treatment means differ. As with all hypothesis tests, the test is conducted under the assumption that the null hypothesis, EMBED Equation.DSMT4 , is correct. Assuming the errors are iid EMBED Equation.DSMT4 , the following are all true under the null hypothesis (Note: n is the total size of the sample):  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Under the null hypothesis, the following are also true:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  So, under the null hypothesis of the test we expect the F - Ratio to be about 1. Clearly, an F - Ratio very different from one is evidence for the alternative hypothesis. The question is: is the test left-, right-, or two-tailed. The answer is: it's right-tailed. The reason is that under the alternative hypothesis, EMBED Equation.DSMT4 , the following are true:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Notice that under EMBED Equation.DSMT4 the expected mean square for the model, E{MSR}, is greater than EMBED Equation.DSMT4 , so the F - Ratio is expected to be greater than 1. Hence, we reject EMBED Equation.DSMT4 in favor of EMBED Equation.DSMT4 , i.e., we conclude there is a differential treatment effect, for values of F significantly greater than 1. Although we could use a table of critical values of the F distribution with EMBED Equation.DSMT4 and EMBED Equation.DSMT4 to conduct the test at a fixed significance level  EMBED Equation.DSMT4 , we'll rely on the P-value in the ANOVA table of computer output. Notice that EMBED Equation.DSMT4 under the alternative hypothesis. So we are more likely to be able to reject the null hypothesis (and conclude a treatment effect exists) if either of the following are true: At least one of the  EMBED Equation.DSMT4  is large, i.e., the larger the treatment effect is, the more likely it is that we'll detect it! The  EMBED Equation.DSMT4  are large. The larger the samples are, the more precisely we can estimate the treatment effects EMBED Equation.DSMT4  (making it easier to detect differences between them). Analysis of Factor Level Effects Suppose that the errors appear to be independent EMBED Equation.DSMT4  based upon an analysis of the residuals, and that the P-value for the model leads us to conclude that some of the treatment means differ. The next step is to investigate the nature of the differences in treatment effects. From your introductory statistics course, you may be familiar with the t-test for the difference in two population means,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Although this is not the test that is usually used in the analysis of variance, it is a good place to start a discussion about analyzing treatment effects in one-way ANOVA. A t-Test for the Difference in the Means of Two Treatments Suppose that our interest lay only in considering whether the mean value of the dependent variable is different for two different levels of the factor variable. If the assumption that the errors are iid  EMBED Equation.DSMT4  appears reasonable, then we can construct confidence intervals, and conduct hypothesis tests, for the difference in the means,  EMBED Equation.DSMT4 . Confidence intervals and hypothesis tests are most straightforward when they involve a single parameter estimated by a single random variable! (I wish I could get my Stat 50 students to realize this.) So we should think of the difference in the means,  EMBED Equation.DSMT4 , as a single parameter. Then the obvious choice for an estimator is the random variable  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  are the means of independent samples drawn from the two treatments. The following notation summarizes where we are at this point.  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are the sizes of the samples drawn from treatment populations one and two, respectively.  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , the variances for the dependent variable in each population, are assumed equal, i.e.,  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4 is simply the variance of the error.  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 are the sample means under the two treatments. Now, any confidence interval for, or hypothesis test of,  EMBED Equation.DSMT4 will depend upon the distribution of the estimator  EMBED Equation.DSMT4 . Proceeding step by step,  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4   EMBED Equation.DSMT4 , because linear combinations of normal variables are also normal, and the variance of the difference of two independent random variables is the sum of their individual variances. Although we've assumed constant variance  EMBED Equation.DSMT4 , we've made no assumption about the actual value of  EMBED Equation.DSMT4 . Therefore, we must estimate  EMBED Equation.DSMT4  from the samples taken from the populations. The most efficient way to estimate  EMBED Equation.DSMT4 is by through the pooled estimate  EMBED Equation.DSMT4 , where,  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4 are the usual sample variances for the two samples drawn. Then, from well established theory, the statistic (just another name for a random variable computed from the data)  EMBED Equation.DSMT4  has a t distribution with  EMBED Equation.DSMT4 . Example: Five measurements of the carbon content (in ppm) of silicon wafers were recorded on successive days of production. Can we conclude that the carbon content has changed? Day 12.012.132.202.092.07Day 22.312.412.232.192.26 (The values in the table below were calculated on my calculator) PopulationsSample SizeSample MeanSample VariancePooled VariancedfCritical Value UsedDay 152.100.00500.00618 EMBED Equation.DSMT4 Day 252.280.0072 Note: Because the sample sizes were equal in this example, the pooled variance is just the average of the two sample variances. In general, however, the pooled variance is a weighted average of the sample variances, where greater weight is placed on the estimate derived from the larger sample. This should seem reasonable since larger samples tend to provide more accurate estimates, and therefore should carry more weight in the pooled estimate. Then a 95% confidence interval for the difference in mean carbon content for the two days is given by  EMBED Equation.DSMT4 , or (0.066 ppm, 0.294 ppm). Similarly, we can conduct a two-tailed test of the equality of the mean carbon content for the two days, with hypotheses  EMBED Equation.DSMT4   EMBED Equation.DSMT4  The P-value for this test of 0.00665 (obtained from my calculator) suggests that the mean carbon content of the wafers for the two days differs, a conclusion we could have reached based on the confidence interval for the difference derived above. Now, you're probably wondering why I've spent so much time on a test that rarely gets used in the analysis of variance. The first reason is that t-tests of the difference between the means of two populations are popular in statistics, and this is a class in second semester statistics. The second reason is more important to the analysis of variance: why, if they are so popular, don't we use t-tests in ANOVA to evaluate the differences in treatment means? This answer involves the idea of the family-wise error rate. The Family-Wise Error Rate Suppose that a factor has k levels and that we've concluded, based on the P-value in the ANOVA table, that some level means are unequal. We then set out to investigate the differences with the goal of (hopefully) ranking the treatment means. If we decide to conduct t-tests for every possibly pairing of treatments, we discover that there are  EMBED Equation.DSMT4  ways to do this. The problem is not merely that this could involve conducting a large number of t-tests, but the much less obvious problem of the family-wise error rate for such a strategy. In the two-tailed t-tests being considered here, the significance level of the test,  EMBED Equation.DSMT4 , is the probability of rejecting the statement  EMBED Equation.DSMT4 when the means for the two treatments are equal, i.e., the probability of committing a Type I error. To repeat myself,  EMBED Equation.DSMT4 . But we are conducting not one such test, but  EMBED Equation.DSMT4 such tests. The question is, what is the probability that at least one Type I error occurs for the family of  EMBED Equation.DSMT4 tests. This is called the family-wise error rate, and as we'll see, it can be much larger that  EMBED Equation.DSMT4 . If the  EMBED Equation.DSMT4  tests were independent (they actually aren't because the same data is used in multiple tests), then it's easy to see that P(at least one Type I error) = 1 - P(no Type I errors) =  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4 is the number of pairings. In fact, this forms an (approximate) lower bound on the family-wise rate, while an upper bound is given by  EMBED Equation.DSMT4 . For  EMBED Equation.DSMT4 , the table below summarizes the family-wise error rates for a few choices of k. Number of Levels, kp"Lower Bound" on P(at least one Type I error)Upper Bound on P(at least one Type I error)33 EMBED Equation.DSMT4 0.14 EMBED Equation.DSMT4 0.1546 EMBED Equation.DSMT4 0.26 EMBED Equation.DSMT4 0.30510 EMBED Equation.DSMT4 0.40 EMBED Equation.DSMT4 0.50615 EMBED Equation.DSMT4 0.54 EMBED Equation.DSMT4 0.75 Alternatives to All Possible Pairwise t-Tests The problem of family-wise error rates has attracted the attention of some of the biggest names in statistics, who have developed procedures for constructing simultaneous confidence intervals that can also be used to conduct pairwise tests of treatment means. They can be accessed in StatGraphics in the Means Plot window by clicking the right mouse button and selecting Pane Options. Fisher's Least Significant Difference (LSD) intervals: Named after Sir R. A. Fisher, this is the method that StatGraphics defaults to (it appears in some of my solutions for no better reason than this). Tukey's Honest Significant Difference (HSD) intervals: Named for John Tukey, who worked for AT&T back when it had more money than God (and better service), this method was specifically designed to control the family-wise Type I error rate for all possible pairwise comparisons of treatment means at a fixed  EMBED Equation.DSMT4 . In most cases, this is the set of intervals that are preferred. Scheffe Intervals: Named for Henry Scheffe, who, besides deriving his intervals, wrote a classic text on the analysis of variance. This procedure was about more than just confidence intervals and pairwise comparisons. It was designed for the related problem of drawing inference on contrasts. We haven't discussed contrasts yet, and we may not have time to this semester, but Scheffe came up with a way to conduct tests of all possible contrasts at a fixed family-wise rate  EMBED Equation.DSMT4 . Scheffe intervals, however, tend to be more conservative, i.e., wider, than HSD intervals. Bonferroni Intervals: One of the original attempts at solving the problem of family-wise error rates, Bonferroni intervals are still useful in certain situations. Generally, however, Tukey's HSD intervals are probably those most commonly employed to draw simultaneous inference in ANOVA at a fixed  EMBED Equation.DSMT4 . Example: (This is the first example explored in the original notes on the analysis of variance.) As city manager, one of your responsibilities is purchasing. The city is looking to buy lightbulbs for the citys streetlights. Aware that some brands lightbulbs might outlive other brands lightbulbs, you decide to conduct an experiment. Seven lightbulbs each are purchased from four brands (GE, Dot, West, and a generic) and placed in streetlights. The lifetime of each of the 28 lightbulbs is then recorded in the file Lightbulbs. Let's consider four different 95% confidence intervals for the difference of the means of the GE and Dot lightbulbs. (Actually, I'm only interested in the width of the intervals since they will all be centered about the same point estimate of the difference in the means.) The table below contains all of the relevant statistics for the four intervals to be created. BrandSample SizeSample MeanSample VarianceGE72.3360.0460Dot72.00.0213West71.7870.0152generic72.10.0105 First we will construct a simple t-interval based solely on the samples taken from the GE and Dot populations. For  EMBED Equation.DSMT4 , the pooled estimate of the variance is  EMBED Equation.DSMT4 . The 95% confidence interval for  EMBED Equation.DSMT4 is  EMBED Equation.DSMT4  =  EMBED Equation.DSMT4   EMBED Equation.DSMT4  EMBED Equation.DSMT4 . The t-interval for  EMBED Equation.DSMT4 using Fisher's LSD is similar to the interval above, but with the pooled estimate of the variance and the degrees of freedom derived from the MSE estimate of the error variance  EMBED Equation.DSMT4 computed from the ANOVA table. The general form for a Fisher EMBED Equation.DSMT4 LSD confidence interval for  EMBED Equation.DSMT4  is  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4 is the degrees of freedom associated the error sum of squares SSE. For the light bulb data, these are MSE = 0.02324 and df = 24. The 95% Fisher t-interval becomes  EMBED Equation.DSMT4  The t-interval for  EMBED Equation.DSMT4 using Bonferroni's method is similar to the LSD interval above, but replacing  EMBED Equation.DSMT4 with  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  is the number of pairwise comparisons. A Bonferroni 95% family-wise t-interval for  EMBED Equation.DSMT4 is given by  EMBED Equation.DSMT4   EMBED Equation.DSMT4 . Tukey's HSD intervals use a critical value drawn from a Studentized Range distribution with  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  (compare this with the F-test in one-way analysis of variance where  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 ). Tables for the Studentized Range distribution appear in statistics texts, or can be found online. The critical value for the Studentized Range is written  EMBED Equation.DSMT4 . A Tukey 95% family-wise confidence interval for  EMBED Equation.DSMT4 is given by  EMBED Equation.DSMT4  The Bonferroni interval is wider than Fisher's LSD, but has the advantage of guaranteeing a fixed level of confidence for the family of all pairwise comparisons. Unless you've decided to focus on one particular comparison before gathering the samples, i.e., a-priori, the Bonferroni is better because it guarantees a family-wise error rate. Tukey's interval was narrower than Bonferroni's. This will be the case whenever all pairwise comparisons are considered, as when the choice of comparisons is made after seeing the data (post-hoc). Tukey intervals are the most widely used when multiple pairwise comparisons are envisioned. $% , . 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