Answer to exercise: Growth of guinea pigs

Answer to exercise: Growth of guinea pigs The effect of a vitamin E diet on the growth of guinea pigs is investigated in the following way: In the beginning of week 1, 10 animals received a growth inhibitor. Then followed a randomisation, so that 5 animals received vitamin E therapy from the beginning of week 5, while the other 5 animals served as controls. In the end of the weeks 1, 3, 4, 5, 6 and 7, the weight of each animal was recorded. The data is in the file vitamin.txt on the home page, with variables grp (1: control, 2: vitamin E), animal, week and weight (in g), and with a total of 2*5*6=60 observations. The first line contains the variable names. We wish to describe the effect of vitamin E on the growth of the animals. 1. Read in the data and construct a more informative variable (called group) to denote the groups. DATA vitamin; INFILE "http://publicifsv.sund.ku.dk/~jufo/courses/rm2016/vitamin.txt" URL FIRSTOBS=2; INPUT grp animal week weight; IF grp=1 THEN group= Control ; IF grp=2 THEN group= VitaminE ; 2. Describe the data using suitable plots and summary statistics. First we make spaghetti plots for each group separately: PROC SGPANEL DATA=vitamin; PANELBY group; SERIES X=week Y=weight / GROUP=animal; 1

We supplement with summary statistics and plot the sample means and standard deviations over time for the two groups: PROC MEANS NWAY DATA=vitamin MEAN STDDEV; CLASS group week; VAR weight; OUTPUT OUT=vitameans MEAN=average STDDEV=sd; PROC SGPLOT DATA=vitameans; SERIES x=week y=average / GROUP=group; PROC SGPLOT DATA=vitameans; SERIES x=week y=sd / GROUP=group; 2

Are there any trends in mean and variance over time? On average both groups gain in weight during the first five weeks. From week 5 on there is a tendency that the animals in the control group stop growing, while the animals in the vitamin E group continue to gain in weight. Variance tends to increase over time in both of the groups. How do we expect to see an effect of vitamin E in these profiles? If vitamin E has an effect on growth, it should of course only be there from week 5 and on. Treatment was randomised and both groups were treated similarly untill the beginning of week 5. 3

Since we do not possess any particular theoretical knowledge about growth of guinea pigs, we start out by letting each week have its own mean value, i.e. by treating week as a factor (a class-variable). 3. Fit a mixed model with an unstructured covariance and a two-way ANOVA structure for the mean (group week group*week). We run the analysis of response profiles using the code PROC MIXED DATA=vitamin; CLASS animal week (REF= 1 ) grp (REF= Control ); MODEL weight = week group group*week / SOLUTION CL DDFM=KR; REPEATED week / TYPE=UN SUBJECT=animal R RCORR; from which we get the following output: The Mixed Procedure Model Information Data Set Dependent Variable Covariance Structure Subject Effect Estimation Method Residual Variance Method Fixed Effects SE Method Degrees of Freedom Method WORK.VITAMIN weight Unstructured animal REML None Kenward-Roger Kenward-Roger Class Level Information Class Levels Values animal 10 1 2 3 4 5 6 7 8 9 10 week 6 3 4 5 6 7 1 group 2 VitaminE Control Dimensions Covariance Parameters 21 Columns in X 21 Columns in Z 0 Subjects 10 Max Obs Per Subject 6 Number of Observations Number of Observations Read 60 Number of Observations Used 60 Number of Observations Not Used 0 Iteration Evaluations -2 Res Log Like Criterion 0 1 526.36555889 1 1 437.84710547 0.00000000 The Mixed Procedure Convergence criteria met. Estimated R Matrix for animal 1 Row Col1 Col2 Col3 Col4 Col5 Col6 1 649.05 714.65 453.00 707.85 623.35 742.50 2 714.65 1703.40 1302.30 1903.85 1538.98 2027.50 4

3 453.00 1302.30 1175.45 1623.57 1654.45 1754.13 4 707.85 1903.85 1623.57 2843.40 2441.15 2885.63 5 623.35 1538.98 1654.45 2441.15 3657.15 3191.50 6 742.50 2027.50 1754.13 2885.63 3191.50 3566.75 Estimated R Correlation Matrix for animal 1 Row Col1 Col2 Col3 Col4 Col5 Col6 1 1.0000 0.6797 0.5186 0.5211 0.4046 0.4880 2 0.6797 1.0000 0.9203 0.8651 0.6166 0.8226 3 0.5186 0.9203 1.0000 0.8881 0.7980 0.8567 4 0.5211 0.8651 0.8881 1.0000 0.7570 0.9061 5 0.4046 0.6166 0.7980 0.7570 1.0000 0.8837 6 0.4880 0.8226 0.8567 0.9061 0.8837 1.0000 Covariance Parameter Estimates Cov Parm Subject Estimate UN(1,1) animal 1703.40 UN(2,1) animal 1302.30 UN(2,2) animal 1175.45 UN(3,1) animal 1903.85 UN(3,2) animal 1623.57 UN(3,3) animal 2843.40 UN(4,1) animal 1538.98 UN(4,2) animal 1654.45 UN(4,3) animal 2441.15 UN(4,4) animal 3657.15 UN(5,1) animal 2027.50 UN(5,2) animal 1754.13 UN(5,3) animal 2885.63 UN(5,4) animal 3191.50 UN(5,5) animal 3566.75 UN(6,1) animal 714.65 UN(6,2) animal 453.00 UN(6,3) animal 707.85 UN(6,4) animal 623.35 UN(6,5) animal 742.50 UN(6,6) animal 649.05 Standard Effect group week Estimate Error DF t Value Pr > t Alpha Intercept 466.40 11.3934 8 40.94 <.0001 0.05 week 3 53.0000 13.5878 8 3.90 0.0045 0.05 week 4 102.40 13.5536 8 7.56 <.0001 0.05 week 5 95.2000 20.3801 8 4.67 0.0016 0.05 week 6 80.2000 24.7366 8 3.24 0.0118 0.05 week 7 105.60 23.3701 8 4.52 0.0020 0.05 week 1 0..... group VitaminE 28.0000 16.1127 8 1.74 0.1205 0.05 group Control 0..... week*group VitaminE 3 3.6000 19.2161 8 0.19 0.8561 0.05 week*group Control 3 0..... week*group VitaminE 4-22.6000 19.1677 8-1.18 0.2722 0.05 week*group Control 4 0..... week*group VitaminE 5-22.6000 28.8219 8-0.78 0.4556 0.05 week*group Control 5 0..... week*group VitaminE 6 28.4000 34.9829 8 0.81 0.4404 0.05 week*group Control 6 0..... week*group VitaminE 7 44.0000 33.0503 8 1.33 0.2198 0.05 week*group Control 7 0..... week*group VitaminE 1 0..... week*group Control 1 0..... Effect group week Lower Upper Intercept 440.13 492.67 week 3 21.6664 84.3336 week 4 71.1454 133.65 week 5 48.2034 142.20 week 6 23.1573 137.24 week 7 51.7086 159.49 week 1.. group VitaminE -9.1560 65.1560 group Control.. week*group VitaminE 3-40.7125 47.9125 week*group Control 3.. week*group VitaminE 4-66.8007 21.6007 week*group Control 4.. week*group VitaminE 5-89.0633 43.8633 week*group Control 5.. week*group VitaminE 6-52.2706 109.07 5

week*group Control 6.. week*group VitaminE 7-32.2141 120.21 week*group Control 7.. week*group VitaminE 1.. week*group Control 1.. Is there evidence of differences in growth between the groups? The P-value for the interaction is 0.1841, so there is no evidence of overall differences in growth pattern between the two groups. The 5 estimates for week indicate the predicted weigth gains since baseline for the control group (the reference group), whereas the interaction term estimates indicate the differences in weight gain between the vitamin group and the control group group. We don t see a consistent pattern in the differences betwen the two groups, but at the last two follow-ups the vitamin group has on average gained more weight than the controls. Give an estimate of the difference in weight gain between the groups at the end of week 7 (with 95% confidence limits). We expect a more powerful test would be to compare the weigth gains at final follow-up where the difference between the groups ought to be the largest. The estimated difference in weight gain after seven weeks is 44.0 g with a confidence interval of -32.21 to 120.21 g. Although the estimate is in favour of vitamin E treatment, the effect is not significant (P=0.2198) Take a look at the estimated R correlation matrix. Are there any trends in the correlations? We see that the correlations tend to be stronger for observations that are close in time. 4. Should we do baseline adjustment in this investigation? Yes, we should. The two groups are randomized and treated in an identical fashion in the first three time points, so any difference in this period has to be pure coincidence. By taking the known equality of means at the first 3 time points into account, we should be able to obtain a more powerful analysis. 5. Make a dataset consisting only of data from weeks 1 through 4. Make an apropriate analysis to compare the two groups. Is there evidence of differences in means at any time point? We make a new dataset called period1 and run the analysis of response profiles for the first three time points only. To get estimates of the difference in means between the two groups at each time point, we omit the group term from the model specification 6

DATA period1; SET vitamin; IF week < 5; PROC MIXED DATA=period1; CLASS animal week (REF= 1 ) group (REF= Control ); MODEL weight = week group*week / SOLUTION CL DDFM=KR; REPEATED week / TYPE=UN SUBJECT=animal; From this we get the following results (initial output omitted): Standard Effect group week Estimate Error DF t Value Pr > t Alpha Intercept 466.40 11.3934 8 40.94 <.0001 0.05 week 3 53.0000 13.5879 8 3.90 0.0045 0.05 week 4 102.40 13.5536 8 7.56 <.0001 0.05 week 1 0..... week*group VitaminE 3 31.6000 26.1029 8 1.21 0.2606 0.05 week*group Control 3 0..... week*group VitaminE 4 5.4000 21.6836 8 0.25 0.8096 0.05 week*group Control 4 0..... week*group VitaminE 1 28.0000 16.1127 8 1.74 0.1205 0.05 week*group Control 1 0..... Effect group week Lower Upper Intercept 440.13 492.67 week 3 21.6663 84.3337 week 4 71.1453 133.65 week 1.. week*group VitaminE 3-28.5933 91.7933 week*group Control 3.. week*group VitaminE 4-44.6026 55.4026 week*group Control 4.. week*group VitaminE 1-9.1560 65.1560 week*group Control 1.. Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F week 2 7 48.57 <.0001 week*group 3 6 1.64 0.2768 We find no evidence of differences between the two group (P=0.2768). In fact, we know that any difference between the two groups in weeks 1 through 4 must be pure coincidence, so we need not perform this analysis. In case a significant difference was found this would be a type I error and not a useful result. In what follows we will try to make a more specific model for the mean using the data from the entire study period. 7

6. Add new covariates to the data as suggested below: DATA vitamin; SET vitamin; IF week>4 THEN treat=group; ELSE treat= Control ; IF grp=2 AND week>4 THEN vitaweeks=week-4; ELSE vitaweeks=0; Try to explain in words the content of the two variables. The variable treat describes the treatment that the animal actually received on a week to week basis. In weeks 1 through for the vitamin E group got the same treatment as the controls while in weeks 5 through 7 they received vitamin E. The variable vitaweeks records how many weeks the animal has received vitamin E. That will be zero at all times for the animals in the control group. The animals in the vitamin E groups has had vitamin E for one week at the end of week 5, two weeks at the end of week 6, and three weeks at the end of week 7. We will use vitaweeks as a numerical variable in the analyses to estimate the average weight gain per week with vitamin E. We now use the above defined variables as covariates in various combinations. Try the following model specifications: (a) week treat*week (b) week vitaweeks (c) week vitaweeks treat*week The same program is run with the three different model specifications (to be uncommented one at a time): PROC MIXED DATA=vitamin; CLASS animal week (REF= 1 ) treat (REF= Control ); * MODEL weight = week week*treat / SOLUTION CL DDFM=KR OUTPM=fita; * MODEL weight = week vitaweeks / SOLUTION CL DDFM=KR OUTPM=fitb; * MODEL weight = week vitaweeks week*treat / DDFM=KR OUTPM=fitc; REPEATED week / TYPE=UN SUBJECT=animal R RCORR; 7. Try to make pictures of the predicted weight profiles for the three models. We use the data from OUTPM to plot the estimated weight profiles from the three different model specifications, e.g. using 8

PROC SGPLOT DATA=fita; WHERE animal in (1,10); SERIES X=week Y=pred / GROUP=group; TITLE Model A ; The predicted profiles from models a and c are exactly the same and this is no coincidence since these are merely different ways of phrasing the same model for the mean. In model a/c means are constrained so that they are identical in the two groups in weeks 1, 3, and 4. From week 5 on the means of the two groups are allowed to evolve differently with no further restrictions. Model b is genuinely different from the other two model specifications and it is more restrictive because it assumes that the difference between the two groups can be explained by a constantly increasing difference in weight gain per week (see estimates below). Is it reasonable to say that the difference between the vitamin and the control group increases linearly with time from week 5 on, or is the effect more complicated than that? Model specification c allows us to test model b against model a using a standard F-test. 9

Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F week 5 5.55 26.35 0.0008 vitaweeks 0... week*treat 2 7 5.76 0.0331 This shows that a model with week and vitaweeks is significantly worse at describing the data. So we conclude that there is more to the effect of vitamin than a constantly increasing difference in weight gain. To judge from the estimated profiles this could be due to a delay in treatment effect. We don t see any indication of treatment effect in week 5, while in weeks 6 and 7, the vitamin E group has gained in weight compared to the controls. 8. For each model, give an estimate of the difference in weight between the groups at the end of week 7. Models a and c are essentially the same but specification a gives us understandable estimates of the treatment effect while specification c is only good for testing. We thus look at the output from model a.. Standard Effect treat week Estimate Error DF t Value Pr > t Alpha Intercept 480.40 8.9146 9 53.89 <.0001 0.05 week 3 54.8000 9.0784 9 6.04 0.0002 0.05 week 4 91.1000 9.7894 9 9.31 <.0001 0.05 week 5 90.0640 17.2335 11.4 5.23 0.0002 0.05 week 6 57.4248 22.1748 10.2 2.59 0.0265 0.05 week 7 99.7444 20.6702 11.3 4.83 0.0005 0.05 week 1 0..... week*treat Control 3 0..... week*treat Control 4 0..... week*treat VitaminE 5-12.3279 20.3233 8-0.61 0.5609 0.05 week*treat Control 5 0..... week*treat VitaminE 6 73.9504 24.7341 8 2.99 0.0173 0.05 week*treat Control 6 0..... week*treat VitaminE 7 55.7112 26.1126 8 2.13 0.0654 0.05 week*treat Control 7 0..... week*treat Control 1 0..... Effect treat week Lower Upper Intercept 460.23 500.57 week 3 34.2632 75.3368 week 4 68.9549 113.25 week 5 52.3069 127.82 week 6 8.1756 106.67 week 7 54.3982 145.09 week 1.. week*treat Control 3.. week*treat Control 4.. week*treat VitaminE 5-59.1934 34.5375 week*treat Control 5.. week*treat VitaminE 6 16.9137 130.99 week*treat Control 6.. week*treat VitaminE 7-4.5045 115.93 week*treat Control 7.. week*treat Control 1.. The estimated difference in weight gains at final follow-up is 55.7g (95% CI: -4.5 to 115.9) in favour of vitamin E. From model b we get the following estimates: 10

Standard Effect week Estimate Error DF t Value Pr > t Alpha Lower Upper Intercept 480.40 8.9147 9 53.89 <.0001 0.05 460.23 500.57 week 3 54.8000 9.0784 9 6.04 0.0002 0.05 34.2632 75.3368 week 4 91.1000 9.7894 9 9.31 <.0001 0.05 68.9549 113.25 week 5 81.9713 14.8428 9.23 5.52 0.0003 0.05 48.5209 115.42 week 6 90.5427 18.6790 9.89 4.85 0.0007 0.05 48.8625 132.22 week 7 121.81 20.4534 9.66 5.96 0.0002 0.05 76.0220 167.61 week 1 0....... vitaweeks 3.8573 7.9195 8 0.49 0.6393 0.05-14.4049 22.1196 This estimates the effect of vitamin E to be an additonal weght gain of 3.86 g per week (95% CI: -14.40 to 22.12). Over the entire study which includes three weeks of vitamin E we would expect a total additional weight gain of 3 3.86 = 11.6 g with a confidence interval of (3 14.40; 3 22.12) = ( 43.2; 66.4). 9. From your preferred model, do you get evidence of difference in growth between the groups? We have evidence that model b does not fit the data as well as model a. Therefore we base our conclusion on model a. There appears to be an additional weight gain with vitamin E at final follow-up but not reaching the level of significance (P=0.0654). Also the overall test of difference in growth pattern between the groups is non-significant (P=0.0945). 10. Investigate whether a compound symmetry model could be used as in place of the unstructured covariance. Does the choice of covariance structure affect the conclusions about the treatment effect? We refit model a with a compound symmetry structure. This results in estimated model parameters and goodnes of fit as follows: Estimated R Matrix for animal 1 Row Col1 Col2 Col3 Col4 Col5 Col6 1 2196.18 1510.69 1510.69 1510.69 1510.69 1510.69 2 1510.69 2196.18 1510.69 1510.69 1510.69 1510.69 3 1510.69 1510.69 2196.18 1510.69 1510.69 1510.69 4 1510.69 1510.69 1510.69 2196.18 1510.69 1510.69 5 1510.69 1510.69 1510.69 1510.69 2196.18 1510.69 6 1510.69 1510.69 1510.69 1510.69 1510.69 2196.18 Estimated R Correlation Matrix for animal 1 Row Col1 Col2 Col3 Col4 Col5 Col6 1 1.0000 0.6879 0.6879 0.6879 0.6879 0.6879 2 0.6879 1.0000 0.6879 0.6879 0.6879 0.6879 3 0.6879 0.6879 1.0000 0.6879 0.6879 0.6879 4 0.6879 0.6879 0.6879 1.0000 0.6879 0.6879 5 0.6879 0.6879 0.6879 0.6879 1.0000 0.6879 6 0.6879 0.6879 0.6879 0.6879 0.6879 1.0000 Covariance Parameter Estimates Cov Parm Subject Estimate CS animal 1510.69 Residual 685.49 Fit Statistics -2 Res Log Likelihood 517.6 AIC (smaller is better) 521.6 11

AICC (smaller is better) 521.9 BIC (smaller is better) 522.2 Standard Effect treat week Estimate Error DF t Value Pr > t Alpha Intercept 480.40 14.8195 15.9 32.42 <.0001 0.05 week 3 54.8000 11.7089 42 4.68 <.0001 0.05 week 4 91.1000 11.7089 42 7.78 <.0001 0.05 week 5 90.6100 15.0639 43.1 6.02 <.0001 0.05 week 6 75.6100 15.0639 43.1 5.02 <.0001 0.05 week 7 101.01 15.0639 43.1 6.71 <.0001 0.05 week 1 0..... week*treat Control 3 0..... week*treat Control 4 0..... week*treat VitaminE 5-13.4201 18.9550 44.6-0.71 0.4826 0.05 week*treat Control 5 0..... week*treat VitaminE 6 37.5799 18.9550 44.6 1.98 0.0536 0.05 week*treat Control 6 0..... week*treat VitaminE 7 53.1799 18.9550 44.6 2.81 0.0074 0.05 week*treat Control 7 0..... week*treat Control 1 0..... Effect treat week Lower Upper Intercept 448.96 511.84 week 3 31.1713 78.4287 week 4 67.4713 114.73 week 5 60.2325 120.99 week 6 45.2325 105.99 week 7 70.6325 131.39 week 1.. week*treat Control 3.. week*treat Control 4.. week*treat VitaminE 5-51.6062 24.7660 week*treat Control 5.. week*treat VitaminE 6-0.6062 75.7660 week*treat Control 6.. week*treat VitaminE 7 14.9938 91.3660 week*treat Control 7.. week*treat Control 1.. Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F week 5 42 27.76 <.0001 week*treat 3 43.7 4.20 0.0107 First we look at the -2 loglikelihood value 517.6 which has to be compared to a similar value of 464.2 in the similar model with the unstructured covariance (output not shown). The CS covariance only has two parameters while the UN covaraince has 6 + 6 (6 1) = 21 parameters. We look up 517.6 464.2 = 53.4 in 2 the Chi-square distribution with 19 degrees of freedom. The test is highly significant (P<0.0001), so the compound symmetry covariance does not fit the data at all well. Really this is no surprise as the variance in the data increases with time and the correlations declines with increasing time span between observations. Turning to the estimates and tests, we see that the compound symmetry model would reach a much stronger conclusion about the effect of vitamin E than is supported by the data; Roughly speaking the compound symmetry structure assumes that the correlation between first and last observation is stronger than it actually is and thereby reaches a too strong conclusion about changes over time. For a final comparison we plot the estimated weight profiles from the two different covariance pattern models first for the controls, next for the vitamin E group. We include confidence intervals in the plots. 12

The most prominent difference between the two model fits is that the confidence intervals from the unstructured covariance model are more narrow in the beginning and wider in the end than those obtained from the compound symmetry model. Also the estimated profiles differ slightly between the models due to a different weighting of the data. 13