1. (Rao example 11.15) A study measures oxygen demand (y) (on a log scale) and five explanatory variables (see below). Data are available as

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1 ST 51, Summer, Dr. Jason A. Osborne Homework assignment # - Solutions 1. (Rao example 11.15) A study measures oxygen demand (y) (on a log scale) and five explanatory variables (see below). Data are available as x 1 : biol. O demand x : total Kjeldahl Nitrogen x 3 : total solids x 4 : total volatile solids x 5 : chem. O demand (a) Among x 1 through x 5, which variables exhibit a significant linear association with y? x 1 (r =.78,p <.0001),x 3 (r =.84,p <.0001),x 4 (r =.71,p =.0004),x 5 (r =.83,p <.0001) (b) Fit a full multiple linear regression model which includes all of the variables you identified in part (a). For each partial regression coefficient, report the p-value for a test that the coefficient is 0. (The problem is even worse if you fit the model with all five of x 1 x 5. Watch what happens to these partial slope p-values in this case.) p-values for partial slopes are in the right-most column below: Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept x x x x (c) Use an F-test to compare the nested (reduced) model µ = β 0 +β x +β 4 x 4 with the full model with all five predictors. From the output below, it can be seen that R(β 1,β 3,β 5 β 0,β,β 4 )/3 = Division by MS[E] f = leads to a significant F-ratio: (F = 7.19,p =.0037,df = 3,14. So, the reduced model may be rejected in favor of the full model. That is, there is significant evidence of dependence on at least of of the variables x 1,x 3,x 5 after accounting for the effects of x and x 4. Test x1x3x5ns Results for Dependent Variable y Mean Source DF Square F Value Pr > F Numerator Denominator (d) Consider the model involving all five predictors, x 1,x,x 3,x 4,x 5. How many subsets with at least one predictor are possible? (Answer: a lot!): ( ) ( ) ( ) ( ) ( ) = (e) Use the C p criterion (or any other reasonable model selection criteria) to choose the best subset model for predicting log-oxygen demand: proc reg; model y=x1-x5/selection=cp; run;

2 C(p) Selection Method Number in Model C(p) R-Square Variables in Model x3 x x x3 x x3 x4 x x1 x3 x x x3 x4 x x1 x x3 x x1 x3 x4 x x4 x x x4 x5 Looks like µ(x 3,x 5 ) = β 0 + β 3 x 3 + β 5 x 5 is preferred. (f) Consider the model µ = β 0 + β 3 x 3 + β 5 x 5. i. Estimate the mean log-oxygen demand when x 3 = 5 and x 5 = 6. I used the following code: proc glm; model y=x3 x5; estimate "problem 1 (f) i." intercept 1 x3 5 x5 6; estimate "problem 1 (g) " x3 1 x5-1; run; to get the following output, which gives an estimated mean log leafburn time of ˆµ(x 3 = 5,x 5 = 6) = 0.3(ŜE = 0.07) Standard Parameter Estimate Error t Value Pr > t problem 1 (f) i problem 1 (g) Report a standard error. Give the product of vectors and matrices that is evaluated to get this standard error. ˆµ = (1,5,6)ˆβ,(vector product) ŜE = (1,5,6)(X X) 1 (1,5,6) MS[E] = ii. Estimate the standard deviation of log-oxygen demand for x 3 = 5 and x 5 = 6. MS[E] = 0.5 (g) Estimate the difference between the slope for x 3 and that for x 5. ˆβ 1 ˆβ 3 = (0,1, 1)ˆβ = 0.007(SE =.1) (See output above. Looks like slopes are plausibly equal.) (h) Fit a simple linear regression model with x 3 + x 5 as the single predictor. Estimate the standard deviation of log-oxygen demands when x 3 +x 5 = 11. Compare with earlier question. MS[E] = 0.4(df = 18), which is smaller than for the more complex model. (i) For a similar challenge, see the NFL problem on the course website (and compare the predicted outcome for the Colts-Patriots game with the observed outcome).. Rao 1.5: (Refer to plantht1.dat and Example 8. in Rao) Four randomly selected seedlings were grown under t = 5 experimental conditions and heights at four weeks were measured:

3 t Label Description Sample mean Sample variance 1 D Darkness AL safelight type A, low intensity AH safelight type A, high intensity BL safelight type B, low intensity BH safelight type B, high intensity (a) Write a general linear model using dummy variables. for seedling i. Assume E i iid N(0,σ ). Y i = β 0 + β 1 x i,al + β x i,ah + β 3 x i,bl + β 4 x i,bh + E i (b) Write a general linear model using factorial effects. Y ij = µ + τ i + E ij for seedling j, light treatment i. Assume E ij iid N(0,σ ) and also that τ i = 0 (c) Conduct an F-test for the null hypothesis that none of the treatments have any effect on mean plant height. H 0 : τ i 0, H 1 : τ i not all0 F = 9.4,p =.0005,df = 4,15 So we reject H 0 and conclude that the light treatments cause height differences in seedlings. (d) Among the non-darkness treatments, express the mean difference between low and high intensity as a function of parameters in part (a) above. Also, do this for part (b). Report an estimate of this effect, along with a standard error. This questions asks about the effect of light intensity. What function of parameters is estimated by the statistic ˆθ = ȳ3+ + ȳ 5+ Substituting in the models above leads to θ = β 3 + β 5 θ = τ 3 + τ 5 ȳ+ + ȳ 4+ β + β 4 (MLR model τ + τ 4 (factorial effects model) Observed value ˆθ = ,ŜE = 0.5. (The technical term for this quantity is the main effect of light intensity on mean plant height. ) 3. Consider the sample of n F = 18 girls and n M = boys in Bigclass.txt as a random sample from a population of interest. Here is the code I used to answer the questions below: proc reg data=kidz; model height = boy; model height = age; model height = age boy; model height = age boy agexboy; run; (a) Use regression with an indicator variable to conduct an equal variances t-test of the hypothesis the average heights of the two populations (boys and girls) are equal. Is this hypothesis plausible in light of these data? Also, do this with software for a two-sample comparisons of means like PROC TTEST and compare the results. How is the pooled sample variance from the two-sample comparison of means related to the error mean square from the regression? Model Y i = β 0 +

4 β 1 x i,boys + E i where x i,boys is an indicator variable taking the value 1 if person i is a boy, 0 otherwise. The sample mean for girls is ˆβ 0 = for boys it is ˆβ 0 + β 1 = for difference of ˆβ 1 = 3.0,SE = 1.8,t =.37,p =.03,df = 38. There is some evidence that in th population from which these children were sampled, the boys are taller than the girls, on average. The pooled variance, S p from the t-test is the same as MS[E] from the regression approach. (b) Conduct a linear regression of height on age, ignoring gender. Is there a significant linear association between height and age? Report a p-value. Yes, the F-ratio is highly significant, indicating evidence of a linear relationship between height and age. (F =.3, p <.0001, df = 1, 38) (c) Fit a model to test the hypothesis of equal mean heights for boys and girls of the same age when assuming the same dependence of height on age for boys as for girls Model is µ(age,x boy ) = β 0 + β 1 age + β x boy, fitting model give ˆβ =.4,t =.35,p =.040, still significant. (d) Test that this dependence on age is constant across the two genders. Model is µ(age,x boy ) = β 0 + β 1 age + β x boy + β 3 age x boy, fitting model give ˆβ 3 = 1.,t = 1.75,p = At level α =.05, this non-additivity coefficient is not significant, suggesting that the hypothesis that the height-age relationship is plausibly the same for boys as for girls.

5 4. Rao 1.3 (1.3a) Construct the one-way ANOVA table for comparing the three treatment means when z is ignored. Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total (a) Conduct an F-test for equality of means. That is, specify a model and a null hypothesis for no therapy effect, then compute the F-ratio F = MS(trt)/MS(E), df =,7 and compare it to the critical value F(.05,,7). Model is Y ij = µ + τ i + E ij, null hypothesis is H 0 : τ i 0. Observed F-ratio is F = 6.71 on df =,7. Critical value is F(.05,,7) = 3.35, so H 0 is rejected at level α =.05 and we have evidence of a therapy effect. (b) Consider the following equivalent model, that leads to the same inference regarding treatment effects,(ignored) (c) Plot y versus z with a different symbol for each therapy. (d) Using PROC GLM or PROC REG, fit the following analysis of covariance (ANCOVA) model: Y i = β 0 + β 1 x i1 + β 1 x i + βz i + E i where X i1 and x i are indicator variables for therapies 1 and respectively. Report each regression coefficient along with a standard error. ˆβ = (1.8,17.1,.85,1.13) and standard errors as given in output: Standard Parameter Estimate Error t Value Pr > t Intercept B trt B <.0001 trt B trt B... z <.0001

6 (e) Report the F-test for a therapy effect, after controlling for the effect of the pretreatment (z) score. F = R(β 1,β β 0,β)/ MS(E) (f) Report the unadjusted post-test score for therapy. Using factorial notation y ij, ȳ + = = 70.1(df =,6) (g) Report the adjusted post-test score for therapy, along with a standard error. Using regression notation ˆβ 0 + ˆβ + ˆβ z = 35.8(ŜE = 1.1)