1 Tomato yield example.

Transcription

1 ST706 - Linear Models II. Spring 2013 Two-way Analysis of Variance examples. Here we illustrate what happens analyzing two way data in proc glm in SAS. Similar issues come up with other software where we need to worry about balance, possible weighting in defining main effects, etc.. In SAS the type I analysis use hierarchical sums of squares (test for that effect accounting only for effects above it in the table) while the type III sums of squares test for an effect accounting for all other things in the model. Note that the default in SAS is to assume that the weights used to define marginal means are equal and so effects add to 0; e.g., i α i = 0. If we want to test for equal marginal means with other weighting it needs to be customized, as illustrated below. Assume constant variance throughout; but, in practice would be evaluated and remedied as needed. 1 Tomato yield example. From Devore s book Probability and Statistics for Engineering and the Sciences and took text describing the experiment from a Duke web site. Three different varieties of tomato (Harvester, Pusa Early Dwarf, and Ife No. 1) and four different plant densities (10, 20, 30 and 40 thousand plants per hectare) are being considered for planting in a particular region. The goal of the experiment is to determine whether variety and density affect yield ( Effects of Plant Density on Tomato Yields in Western Nigeria, Experimental Agriculture, 1976: 43:47). Both factors are randomized, where the unit here is a plot of land. There are 36 such plots and the assumption is that the treatment combinations are randomized to these 36 subject to the constraint that there are 3 plots for each of the 12 treatment combinations. This is actually a setting where the marginal means are probably not of interest if there is interaction, but we use it for illustrating the computational issues nonetheless. Analysis of the tomato yield data, balanced. Illustrates balanced data with equal weights used in defining the marginal and overall means. This is a pretty straightforward analysis. Analysis Variable : Yield Variety Density N Mean Std Dev Minimum Maximum H Ife P *** SOME OF THE SAS CODE *** /* Two-way analysis with inferences for main effects */ proc glm data=a; class density variety; model yield = density variety density*variety; lsmeans density variety/cl pdiff tdiff; lsmeans density variety/cl pdiff tdiff adjust=bon; lsmeans density variety/cl pdiff tdiff adjust=tukey; 1

2 lsmeans density variety/cl pdiff tdiff adjust=scheffe; *** TWO WAY ANALYSIS. **** Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total Source DF Type I SS Mean Square F Value Pr > F Density <.0001 Variety <.0001 Density*Variety Source DF Type III SS Mean Square F Value Density Variety Density*Variety ***BECAUSE OF BALANCED DATA THE TYPE I AND TYPE III SUM OF SQUARES ARE THE SAME.*** ****ONE-AT-A-TIME INFERENCES Density Yield LSMEAN 95% Confidence Limits Difference Between 95% Confidence Limits for LSMEAN Variety Yield LSMEAN Number H Ife P Variety Yield LSMEAN 95% Confidence Limits H Ife P Difference Between 95% Confidence Limits for Adjustment for Multiple Comparisons: ****BONFERONNI ****** Least Squares Means for Effect Density 2

3 Between Confidence Limits for Between Confidence Limits for Adjustment for Multiple Comparisons: *******TUKEY******* Least Squares Means for Effect Density Between Confidence Limits for Least Squares Means for Effect Variety Between Confidence Limits for Adjustment for Multiple Comparisons: ******SCHEFFE ***** Between Confidence Limits for Least Squares Means for Effect Variety Between Confidence Limits for With balanced data means and lsmeans will give the same answer. MORE ANALYSIS OF TOMATO YIELD DATA 3

4 Using the original balanced data. Consider estimating specific linear combinations directly. Since the parameterization is in terms of effects in general we need to write c ij µ ij = µc.. + c i. α i + c.j τ j + c ij γ ij. ij i j ij Examples: 1. µ i. = µ + α i has a coefficient of 1 for µ (intercept) and 1 for α i 2. µ.j = i v iµ ij = µ + i v iα i + τ j + i v iγ ij Illustration: µ.j =.3µ 1j +.2µ 2j +.25µ 3j +.25µ 4j (v 1 =.3, v 2 =.2, etc.) = marginal mean of variety j = expected yield if planted in 30% at density 10000, 20% at density 20000, etc. 3. µ ij has coefficient 1 for µ,α i, τ j and γ ij. How to test a general hypothesis about a set of contrast? Consider testing H 0 : µ.1 = µ.2 = µ.3. This is equivalent to H 0 : µ.1 µ.2 = 0 and µ.1 µ.3 = 0. This is a hypothesis that two linear combinations equal zero. (There are many other sets of two linearly independent contrasts that can be used, but the test is independent of which two we choose.). This is used to show how to easily construct a test for equal marginal means in the presence of interaction when unequal weights are used to define the marginal means and associated main effects. proc glm data=a; class density variety; model yield = density variety density*variety/e clparm; /* ESTIMATE LINEAR COMBINATIONS WITH EQUAL WEIGHTING */ estimate variety1 versus variety3 variety 1 0-1; estimate mean variety 3 intercept 1 variety 0 0 1; estimate mean density 2 variety 3 intercept 1 density variety density*variety ; /* Two-way model with variety marginal means weighted over densities using weights.3,.2,.25 and.25 */ estimate weighted mean for variety1 intercept 1 density variety density*variety ; estimate weighted mean for variety2 intercept 1 density variety density*variety ; estimate weighted mean for variety3 intercept 1 density variety density*variety ; run; /* Showing how to test for equal marginal means with unequal weighting; tests that the specified combinationd equal 0*/ contrast test for variety with unequal weighting variety density*variety , 4

5 variety density*variety ; run; THIS COMES FROM THE e OPTION. IT SHOWS THE FORM OF ESTIMABLE FUNCTIONS ASSUMING THAT EFFECTS ADD TO 0 (SO EQUAL WEIGHTS EVERYWHERE) Effect Intercept General Form of Estimable Functions Coefficients L1 Density L2 Density L3 Density L4 Density L1-L2-L3-L4 Variety H L6 Variety Ife L7 Variety P L1-L6-L7 Density*Variety H Density*Variety Ife Density*Variety P Density*Variety H Density*Variety Ife Density*Variety P Density*Variety H Density*Variety Ife Density*Variety P Density*Variety H Density*Variety Ife Density*Variety P L9 L10 L2-L9-L10 L12 L13 L3-L12-L13 L15 L16 L4-L15-L16 L6-L9-L12-L15 L7-L10-L13-L16 L1-L2-L3-L4-L6-L7+L9+ L10+L12+L13+L15+L16 FROM ESTIMATE STATEMENTS Standard Parameter Estimate Error t Value variety1 versus variety mean variety mean density 2 variety Parameter Pr > t 95% Confidence Limits variety1 versus variety3 < mean variety 3 < mean density 2 variety 3 < Parameter Estimate Error t Value weighted mean for variety weighted mean for variety weighted mean for variety Parameter Pr > t 95% Confidence Limits weighted mean for variety1 < weighted mean for variety2 <

6 weighted mean for variety3 < TEST FROM USE OF THE CONTRAST STATEMENT Contrast DF Contrast SS test for variety with unequal weighting Contrast Mean Square F Value test for variety with unequal weighting Contrast Pr > F test for variety with unequal weighting <.0001 Fitting an additive model using the balanced data. This part fits an additive model. Note that the sums of squares associated with variety and density are the same as in the model with no interaction. This is because of the balance in the data. The test for the main effects is correct no matter what weights are used in defining the main effects (unlike the situation with interaction where the test in the anova table for the main effects in the presence of interaction is only okay if equal weights are involved.) The reason is that since µ ij = µ + α i + τ j, µ.j µ.j = τ j τ j for any weighting (similarly, µ i. µ i. = α i α i ). The consequence here is that the original SS due to interaction (and its associated degrees of freedom) are collapsed into the error term. The Type I and Type III analyses are the same. /* Fitting a model with no interaction */ proc glm data=a; class density variety; model yield = density variety; means density variety; run; Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total Source DF Type III SS Mean Square F Value Pr > F Density <.0001 Variety < Fitting using unbalanced data. Some observations were eliminated to create an unbalanced data set. group N Mean Std Dev 10000H Ife P H Ife P H Ife P H

7 40000Ife P Analysis of model with interaction. The type III analysis is the correct one to use to test for main effects in the presence of interaction, not the type I. (The correctness of the Type III analysis assumes that effects are defined using equal weights; otherwise we need to customize the test as done with the balanced data.) As seen below the Type I sums of squares are the ones that are applicable for a model with no interaction. Notice that the test for interaction is the same in both analyses. Assuming effects are defined with equal weights the lsmeans provides the right analysis of main effects. For example the estimate of the mean at density 1000 is the average of 9.85, and 16.3 which is the given by lsmeans. The means statement just averages all of the values at the given level of a factor. This is a weighted average of the corresponding cell means and would only be appropriate if the marginal means were defined using weights that were proportional to the sample size. proc glm data=au; /* using unbalanced data*/ class density variety; model yield = density variety density*variety; lsmeans density variety/cl pdiff tdiff; means density variety/clm cldiff t; run; Sum of Source DF Squares Mean Square F Value Model Error Corrected Total Source DF Type I SS Mean Square F Value Pr > F Density <.0001 Variety <.0001 Density*Variety Source DF Type III SS Mean Square F Value Density Variety Density*Variety FROM LSMEAN Density Yield LSMEAN 95% Confidence Limits FROM MEANS t Confidence Intervals for Yield Density N Mean 95% Confidence Limits

8 Analysis with no interaction. Type I and Type III SS s agree. These values shows that the Type I sums of squares given above in the ANOVA table with interaction are the ones that apply to an additive model. The analyses of main effects should still be carried out using the lsmeans option (assuming equal weighting). Dependent Variable: Yield Source DF Type I SS Mean Square F Value Density Variety Source DF Type III SS Mean Square F Value Density Variety Senic data example. This data (described in Appendix C.1, see also problem 19.1 of Kutner et al.) is from a sample of hospitals in the U.S. One problem of interest is examining how the infection rates vary with geographical region (A) and whether the hospital has a medical school affiliation or not (B). This is an example where neither factor is experimental. The definition of marginal means and the overall mean are more general than layout in the notes. Suppose in the population the number of hospitals at geographic region i and affiliation j is N ij and the total number of hospitals is N. Define a ij = N ij /N. Then the overall population mean is µ = ij a ijµ ij while the marginal means are given by µ i. = j N ijµ ij /N i. = j w ijµ ij, where w ij = N ij /N i. and µ.j = i v ijµ ij, where v ij = N ij /N.j. So, the weights are proportional to the population sizes. If unknown and we have an overall random sample then we can estimated weights using weights that are proportional to the sample sizes; e.g., w ij = n ij /n i.. Here the use of the means option in SAS produces a correct analysis for the marginal means (estimates and confidence intervals) since if we take a weighted average of cell means using weights proportional to sample size we simply get a mean of all observations over cells that are being combined. In the glm analysis, the test for interaction is okay. It also turns out in this case that the Type I analysis gives the correct test for a main effect in the presence of interaction as long as that main effect is the first one entered into the model. (This is not always true but happens here because the weights are proportional to the sample sizes.) proc glm data=a; class medsch region; model risk = medsch region medsch*region; *lsmeans medsch region/cl; /* Incorrect with weighting*/ means medsch region/clm t; FROM PROC MEANS region medsch N Mean Std Dev region N Mean Std Dev

9 FROM THE GLM ANALYSIS Sum of Source DF Squares Mean Square F Value Model Error Source DF Type I SS Mean Square F Value Pr > F medsch region medsch*region Source DF Type III SS Mean Square F Value Pr > F medsch region medsch*region t Confidence Intervals for risk Error Degrees of Freedom 105 Error Mean Square Critical Value of t % Confidence medsch N Mean Limits % Confidence region N Mean Limits ***ADDITIVE MODEL **** model risk = medsch region; Sum of Source DF Squares Mean Square F Value Model Error Source DF Type I SS Mean Square F Value medsch region Source Pr > F medsch region Source DF Type III SS Mean Square F Value medsch region Source Pr > F medsch region The Type I sums of squares depend on the order of the variables in the model statement. With A = medsch and B = region. SSA = SSE(µ) SSE(µ, α), where SSE(µ) is the error sum of squares for model with just µ in in 9

10 and SSE(µ, α) from the model with µ + α i. SSB = SSE(µ, α) SSE(µ, α, τ). Additional Comments on the senic example. Below is a confidence interval for difference in the two med school levels and simultaneous confidence intervals for the pairwise differences in regions. I ve shown the Bonferroni intervals since they are shorter than Scheffe s. The Tukey intervals are even shorter but with unequal sample sizes Tukey s method (called Tukey-Kramer) is only approximate and performance can be questionable with very unequal sample sizes. The simultaneous pairwise comparisons can be used to carry out a test of equal marginal means across regions. We would reject since we reject that the means for regions 1 and 3 are the same. This also shows how to explicitly carry out a test for equal means across the two medical school categories while allowing interactions using an estimate statement with weights based on the sample sizes. We can use estimate here since there is only one difference. However, constructing the tests this way is not necessary. With weights proportional to sample sizes the estimates of the marginal means are the overall means and the sums of squares for the main effects are the same sums of squares as we would compute for them if viewed as a one-way problem; see page 113 of the notes. This is exactly what gets calculated for the first main effect in the Type I SS. So, the test for medschool from the Type I analysis above is correct with F = 6.61 and a p-value of.0115 agrees with the t-test below, any difference being due to rounding in the weighting. proc glm data=a; class medsch region; model risk = medsch region medsch*region; means medsch/clm cldiff t; means region/clm cldiff bon scheffe tukey; estimate test for medsch with unequal weighting medsch 1-1 region medsch*region ; t Tests (LSD) for risk Alpha 0.05 Error Degrees of Freedom 105 Error Mean Square Critical Value of t Difference medsch Between 95% Confidence Comparison Means Limits *** *** Bonferroni (Dunn) t Tests for risk Difference Simultaneous region Between 95% Confidence Comparison Means Limits ***

11 *** Parameter test for medsch with unequal weighting Standard Estimate Error t Value Pr > t Similarly one could construct an overall F-test for equal means across regions using contrasts. In this case three contrasts are needed since there are four regions; for example with region as factor B you can use µ.1 µ.2, µ.1 µ.3 and µ.1 µ.4. But, as with testing for medschool, you don t need to do this in this case. The analysis above for regions is not correct but we can get the correct test by running the analysis with region first in the model statement and using the Type I analysis. The F-test with a p-value of.0404 below is the correct test for differences among regions. proc glm data=a; class medsch region; model risk = region medsch medsch*region; run; Source DF Type I SS Mean Square F Value P value region medsch medsch*region BIBD example Analysis of BIBD example; martindale wear testing. BIBD EXAMPLE (Martindale wear data). title bibd example ; data values; infile wear.dat ; input run type loss; proc glm; class run type; model loss = run type; lsmeans type/stderr cl tdiff; estimate type 1 mean-i intercept 1 type 1; estimate type 1 mean-ii intercept 7 type 7 run /divisor=7; estimate type1-type2 type 1-1; lsmeans type/cl pdiff=all adjust=tukey; run; Dependent Variable: LOSS Sum of Mean Source DF Squares Square F Value Pr > F Model Error Corrected Total R-Square C.V. Root MSE LOSS Mean

12 Source DF Type I SS Mean Square F Value Pr > F RUN TYPE Source DF Type III SS Mean Square F Value Pr > F RUN TYPE **The means option does not produce anything useful here as it uses mean for a level of a factor over the nonempty cells for that level. Instead we use lsmeans** Standard LSMEAN type loss LSMEAN Error Pr > t Number < < < < < < < Least Squares Means for Effect type t for H0: LSMean(i)=LSMean(j) / Pr > t Dependent Variable: loss i/j < <.0001 < <.0001 <.0001 <.0001 < < < < < < <.0001 < < < <.0001 <.0001 <.0001 < < <.0001 type loss LSMEAN 95% Confidence Limits Least Squares Means Adjustment for Multiple Comparisons: Tukey-Kramer Least Squares Means for effect type Pr > t for H0: LSMean(i)=LSMean(j) Dependent Variable: loss i/j

13 <.0001 <.0001 < < < < < < < < <.0001 <.0001 <.0001 < < <.0001 Least Squares Means for Effect type Between Confidence Limits for Standard Parameter Estimate Error t Value Pr > t type 1 mean-i <.0001 type 1 mean-ii <.0001 type1-type <