Answer to exercise: Blood pressure lowering drugs

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1 Answer to exercise: Blood pressure lowering drugs The data set bloodpressure.txt contains data from a cross-over trial, involving three different formulations of a drug for lowering of blood pressure: A: 50 mg tablet B: 100 mg tablet C: Sustained-release formulation capsule A total of 12 male volunteers were randomly divided into three groups (group=1,2,3), and the groups received each of the three formulations, but in different sequences (and with a wash-out period of one week), according to the scheme below: 1. A-B-C 2. B-C-A 3. C-A-B The outcome for assessing the effect of each treatment was the duration of the drug, in hours. 1. Read in the data and make pictures to illustrate the possible effects. Immediately after reading in the data, we redefine group as the new variable sequence to be able to keep track of the order of treatments in each group. Moreover, we calculate some summary statistics to get an impression of the data: data a1; infile " URL firstobs=2; input id group treatment$ period duration; if group=1 then sequence= ABC ; if group=2 then sequence= BCA ; if group=3 then sequence= CAB ; proc means nway N mean stddev data=a1; class sequence treatment period; var duration; output out=average mean=mean_duration; 1

2 which provides us with the output Analysis Variable : duration N sequence treatment period Obs N Mean Std Dev ABC A B C BCA A B C CAB A B C Note the design: For each group=sequence, we have only 3 of the 9 possible combinations of treatment*period. Next, we make two panels of series plots: one where we plot against treatment and one against period, both subdivided according to sequence (group). proc sort data=a1; by sequence id treatment; proc sgpanel noautolegend data=a1; panelby sequence / rows=1; series Y=duration X=treatment / group=id; proc sort data=a1; by sequence id period; proc sgpanel noautolegend data=a1; panelby sequence / rows=1; series Y=duration X=period / group=id; 2

3 There seems to be a reasonable consistent tendency in the treatment effect, but no obvious consistent pattern in the period effect. If we average over individuals, we get proc sort data=average; by sequence period; proc sgplot data=average; series Y=mean_duration X=period / group=sequence; proc sort data=average; by sequence treatment; 3

4 proc sgplot data=average; series Y=mean_duration X=treatment / group=sequence; From the graph on the left hand side, we clearly see an effect of treatment, although the red sequence (BCA) behaves differently from the other two for treatment A (last treatment) where the duration is somewhat prolonged. Could it be a carry-over effect? Treatment C is seen to have the largest effect, and this is perhaps not very surprising due to the nature of this (sustained-release), followed by the high-dose B. The right hand graph is not that clear, probably because the treatment effects obscure the possible effect of period. Of course, we hope that there is no effect of period at all. Perhaps it is clearer, if we do not divide according to sequence, but rather according to either treatment or period, i.e. proc sort data=average; by treatment period; proc sgplot data=average; series Y=mean_duration X=period / group=treatment; 4

5 proc sort data=average; by period treatment; proc sgplot data=average; series Y=mean_duration X=treatment / group=period; Note, however, that in these plots, the lines do not correspond to time effects for any individual, since the groups are mixed up. In the left hand figure, the pattern for period 1 is clearly to say that C is the best treatment, followed by B and then A. The same conclusion holds for the two other periods, although the pattern here is not as clear. In the right hand side picture, we see a tendency for higher values in later periods, corresponding to a sort of cumulative effect over time, i.e. a plausible carry-over effect. 2. Which factors could determine the duration of the drug? And which of these can be considered systematic? And random? We have three factors that may have an influence on the outcome: The treatment, the period and the subject. The two first of these must be 5

6 considered systematic, whereas the subject may be considered random. We might also think of a possible interaction between treatment and period, in the form of a carry-over effect. Actually, this concept is a bit tricky here, with three periods and three treatments, since we may in principle have a different carry-over effect from treatment A to treatment B than e.g. from treatment A to treatment C. To complicate matters, we may even have a different carry-over effect, if a treatment is given after two other treatments. At first, we disregard possible carry-over effects. 3. Test equality of mean durations from the three formulations/treatments, taking into account a possible period effect. Also, take the correlation between measurements on the same individual into account by including a random effect of individual. We can analyze the data in a mixed model, with systematic effect of the factors period and treatment, and a random effect of subject id: proc mixed data=a1; class sequence id period treatment; model duration = period treatment / ddfm=kr s cl; random intercept / subject=id v vcorr; estimate 100 mg vs 50 mg treatment / cl; which gives us Class Level Information Class Levels Values sequence 3 ABC BCA CAB id period treatment 3 A B C Dimensions Covariance Parameters 2 Columns in X 7 Columns in Z Per Subject 1 Subjects 12 Max Obs Per Subject 3 6

7 Number of Observations Number of Observations Read 36 Number of Observations Used 36 Number of Observations Not Used 0 Estimated V Matrix for id Estimated V Correlation Matrix for id Covariance Parameter Estimates Cov Parm Subject Estimate Intercept id Residual Fit Statistics -2 Res Log Likelihood 49.7 AIC (smaller is better) 53.7 AICC (smaller is better) 54.2 BIC (smaller is better) 54.7 Solution for Fixed Effects Effect treatment period Estimate Error DF t Value Pr > t Intercept <.0001 period period period treatment A <.0001 treatment B treatment C Effect treatment period Alpha Lower Upper Intercept period period period 3... treatment A treatment B treatment C... Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F period treatment <

8 Estimates Label Estimate Error DF t Value Pr > t Alpha 100 mg vs 50 mg Label Lower Upper 100 mg vs 50 mg We note a highly significant difference between the treatments (P < ), but unfortunately also a highly significant increase in duration with time (period, P = ). We also note, that the correlation between measurements taken on the same individual is This correlation is assumed to be the same no matter which two observations we consider, i.e. between all three possible pairs of observations. Note that the analysis above could have been made with a fixed subject effect instead: proc glm data=a1; class id treatment period; model duration = id treatment period / solution; giving instead the output The GLM Procedure Class Level Information Class Levels Values id treatment 3 A B C period Dependent Variable: duration Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE duration Mean Source DF Type III SS Mean Square F Value Pr > F id

9 treatment <.0001 period Parameter Estimate Error t Value Pr > t Intercept B <.0001 id B id B id B id B id B id B id B id B id B id B id B id B... treatment A B <.0001 treatment B B treatment C B... period B period B period B... We note that estimates, standard errors and P-values for treatment and period are exactly the same as in the mixed model above. So why bother with a mixed model? Several reasons actually: Evaluation of cross-over effects, possible covariates for subjects, quantification of subject differences, possible extensions to other covariance patterns, ease of reading output,... (a) What is the estimated difference in duration between 100 mg and 50 mg tablets? Remember to quantify with confidence intervals. Since this particular comparison (which is treatment B vs. A) is not one shown by default, we need either to reparametrize to another reference level, or to make an estimate statement. The latter is the easiest, and it was also included in the code above: estimate 100 mg vs 50 mg treatment / cl; which creates the additional output: Estimates Label Estimate Error DF t Value Pr > t Alpha 100 mg vs 50 mg

10 Label Lower Upper 100 mg vs capsule i.e. an estimated prolonged duration of 0.58 hours, with confidence interval CI=(-1.00, -0.37), P= (b) Do we see any effect of period? Yes, as already noticed, we see a highly significant increase in duration with time (period, P = ), as seen from the pattern repeated below: Effect treatment period Estimate Error DF t Value Pr > t Intercept <.0001 period period period treatment A <.0001 treatment B treatment C Solution for Fixed Effects Effect treatment period Alpha Lower Upper Intercept period period period 3... treatment A treatment B treatment C... Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F period treatment <.0001 These estimates tell us that period 1 gives an expected duration of 0.55 hours less than period 3, whereas period 2 has a duration between these two, and closest to period 3. This could be cautiously interpreted as a sort of cumulative effect over time, i.e. a carry-over effect, just as we saw in a previous picture. 10

11 4. What can we say about the heterogeneity among the subjects? When subjects are treated as fixed effects in the GLM-analysis, we get comparisons of all subjects to the reference subject (the last one in the ad hoc chosen ordering), as well as an overall test for identity of all subjects. This is not interesting, since we do not focus on these particular subjects but rather want to generalize to all subjects (in some specified population). We therefore turn to the mixed model, and use the quantification of between-subject variation: Covariance Parameter Estimates Cov Parm Subject Estimate Intercept id Residual This allows us to calculate typical differences in the duration of a drug (in hours), either between (hypothetical) repetitive measurements of the same treatment in the same period for the same subject, or between measurements of the same treatment in the same period for a different subject: Typical differences (95% Prediction Intervals): for measurements on the same subject ± = ±1.045 for measurements on the different subjects ±2 2 ( ) = ±1.35 The size of these heterogeneity can of course be seen already in the spaghetti plots from question 1, but here, we are able to quantify them. 5. Make two (erroneously) simple tests for the treatment effect, and discuss the possible pitfalls of these analyses: 11

12 A one-way analysis of variance In a one-way anova, we disregard the period effect and we do not take advantage of the pairing (i.e. that all subjects receive all treatments). If the design had been unbalanced, we could get bias due to the confounding between periods and treatments, but in a balanced design such as this, we only loose power. title wrong analysis ; proc glm data=a1; class treatment; model duration = treatment / solution clparm; estimate 100 mg vs 50 mg treatment ; with output wrong analysis The GLM Procedure Dependent Variable: duration Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE duration Mean Source DF Type III SS Mean Square F Value Pr > F treatment <.0001 Parameter Estimate Error t Value Pr > t 100 mg vs 50 mg Parameter 95% Confidence Limits 100 mg vs 50 mg Parameter Estimate Error t Value Pr > t Intercept B <.0001 treatment A B <.0001 treatment B B treatment C B... Note that the estimated difference between treatment A and B is still as in the analyses above, but that the standard error of 12

13 the estimate has increased to instead of the correct level of above. A two-way analysis of variance in treatment and subject. In this analysis, we take the pairing into account, so if there is no period effect, this analysis would be OK. title wrong analysis ; proc glm data=a1; class id treatment; model duration = id treatment / solution clparm; estimate 100 mg vs 50 mg treatment ; with output wrong analysis The GLM Procedure Dependent Variable: duration Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total R-Square Coeff Var Root MSE duration Mean Source DF Type III SS Mean Square F Value Pr > F id treatment <.0001 Parameter Estimate Error t Value Pr > t 100 mg vs 50 mg Parameter 95% Confidence Limits 100 mg vs 50 mg Parameter Estimate Error t Value Pr > t Intercept B <.0001 id B id B id B id B id B id B

14 id B id B id B id B id B id B... treatment A B <.0001 treatment B B treatment C B... Note once more that the estimated difference between treatment A and B is still as in all the previous analyses, and that the standard error is closer to the correct one than in the oneway anova, namely 0.187, pretty close to the one we got from the mixed model analysis (0.151). This reflects the limited effect of the periods (which was not included in this model). We could have produced a another wrong analysis by carrying out a two-way analysis in treatment and period, but this would again not have taken the correlation into account and would therefore be seriously inefficient, just like the one-way ANOVA presented above. 6. With the previously used variance component model, we implicitly assume a compound symmetry covariance structure, but since time/period (and different treatments over time) is involved, this may not be the best choice. Estimate the correlation structure between the three measurements of duration for each subject, by specifying it as unstructured. Does it look like a compound structure? Note, that the use of an unstructured covariance matrix requires the definition of the order of the observations, i.e. the definition of the time scale. It seems most natural to take this to mean periods, but note that this implies that e.g. period 1 and 2 are correlated to the same extent for all subjects, even though this implies different treatment pairs! We specify this by adding period as the repeated factor: proc mixed data=a1; class sequence id period treatment; model duration = period treatment / ddfm=kr s cl; 14

15 repeated period / type=un subject=id r rcorr; and get the 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.A1 duration Unstructured id REML None Kenward-Roger Kenward-Roger Class Level Information Class Levels Values sequence 3 ABC BCA CAB id period treatment 3 A B C Dimensions Covariance Parameters 6 Columns in X 7 Columns in Z 0 Subjects 12 Max Obs Per Subject 3 Number of Observations Number of Observations Read 36 Number of Observations Used 36 Number of Observations Not Used 0 Estimated R Matrix for id Estimated R Correlation Matrix for id Fit Statistics -2 Res Log Likelihood 46.6 AIC (smaller is better)

16 AICC (smaller is better) 62.1 BIC (smaller is better) 61.5 Null Model Likelihood Ratio Test DF Chi-Square Pr > ChiSq Solution for Fixed Effects Effect treatment period Estimate Error DF t Value Pr > t Intercept <.0001 period period period treatment A <.0001 treatment B treatment C Effect treatment period Alpha Lower Upper Intercept period period period 3... treatment A treatment B treatment C... Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F period treatment <.0001 Estimates Label Estimate Error DF t Value Pr > t Alpha 100 mg vs 50 mg Label Lower Upper 100 mg vs 50 mg We note from the above estimate of the correlation structure, that period 2 and 3 are less correlated than the others. But remember, that the sequence of treatments is not the same in the three periods for all individuals. Note also, that the estimates for treatment comparisons have changed, compared to those we got from the CS-analysis in question 3. The estimates for period comparisons, however, remain unchanged. 16

17 We may specify the covariance structure in the order of the treatments instead, i.e. with the code proc mixed data=a1; class sequence id period treatment; model duration = period treatment / ddfm=kr s cl; repeated treatment / type=un subject=id r rcorr; estimate 100 mg vs 50 mg treatment / cl; which yields another output (because it is a different model, specifying a different covariance structure): 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.A1 duration Unstructured id REML None Kenward-Roger Kenward-Roger Class Level Information Class Levels Values sequence 3 ABC BCA CAB id period treatment 3 A B C Dimensions Covariance Parameters 6 Columns in X 7 Columns in Z 0 Subjects 12 Max Obs Per Subject 3 Number of Observations Number of Observations Read 36 Number of Observations Used 36 Number of Observations Not Used 0 Estimated R Matrix for id

18 Estimated R Correlation Matrix for id Fit Statistics -2 Res Log Likelihood 42.1 AIC (smaller is better) 54.1 AICC (smaller is better) 57.6 BIC (smaller is better) 57.0 Solution for Fixed Effects Effect treatment period Estimate Error DF t Value Pr > t Intercept <.0001 period period period treatment A <.0001 treatment B treatment C Effect treatment period Alpha Lower Upper Intercept period period period 3... treatment A treatment B treatment C... Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F period treatment Estimates Label Estimate Error DF t Value Pr > t Alpha 100 mg vs 50 mg Label Lower Upper 100 mg vs 50 mg We note from the above estimate of the correlation structure, that treatment A and B are less correlated than the others. This will be reflected in a larger standard error for this comparison. 18

19 Note also, that the estimates for the period comparisons have now changed, compared to those, we got from the CS-analysis in question 3. The estimates for the treatment comparisons, however, remain unchanged. The two different models above do not agree closely with respect to the estimated effects, and the standard errors also differ, because the assumptions concerning the covariance structure differ. Our predictions from the latter model is shown in the figure below: 19

20 We can summarize our results so far as follows: Number of Model B vs. A -2 log L parameters CS random (0.151) id UN treatment (0.168) order UN period (0.160) order 7. Does the correlation structure seem to depend upon the sequence of administration of the different formulations? We can fit different correlation structures for each sequence group by including group=sequence in the repeated-statement. Note that this gives the same model, no matter whether we specify period or treatment to be the time We sort the data before the fitting to make 20

21 clear how the correlations are to be interpreted. Furthermore, since the covariance structure now depends upon the group, we need to print 3 different structures. We choose to print those of patients number 1, 5 and 9 (one from each group): proc sort data=a1; by sequence id period; title UN period, group=sequence ; proc mixed data=a1; class sequence id period treatment; model duration = period treatment / ddfm=kr s cl; repeated period / type=un subject=id group=sequence r=1,5,9 rcorr=1,5,9; proc sort data=a1; by sequence id treatment; title UN treatment, group=sequence ; proc mixed data=a1; class sequence id period treatment; model duration = period treatment / ddfm=kr s cl; repeated treatment / type=un subject=id group=sequence r=1,5,9 rcorr=1,5,9; The output from the first of these models (with period as time) yields the output (only some of it is shown): UN period, group=sequence Dimensions Covariance Parameters 18 Columns in X 7 Columns in Z 0 Subjects 12 Max Obs Per Subject 3 Estimated R Matrix for id

22 Estimated R Correlation Matrix for id Estimated R Matrix for id Estimated R Correlation Matrix for id Estimated R Matrix for id Estimated R Correlation Matrix for id Fit Statistics -2 Res Log Likelihood 32.6 AIC (smaller is better) 68.6 AICC (smaller is better) BIC (smaller is better) 77.3 Solution for Fixed Effects Effect treatment period Estimate Error DF t Value Pr > t Intercept <.0001 period period period treatment A treatment B treatment C Effect treatment period Alpha Lower Upper Intercept period period period 3... treatment A treatment B treatment C... 22

23 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F period treatment We have to be careful when interpreting the above covariance/correlation estimates, since their meaning depends on the sequence. For id=5, the sequence is BCA, and the correlation matrix therefore tells us, that Corr(A,C)< 0 and that Corr(A,B) is small, but positive. On the other hand, for id=9, the sequence is CAB, and the correlation matrix therefore tells us, that Corr(A,B)< 0. Since the model with treatment as time is the very same model, the only difference in the output will be the order of the elements in the covariance/correlation, and therefore these are the only output, we present: UN treatment, group=sequence The Mixed Procedure Estimated R Matrix for id Estimated R Correlation Matrix for id Estimated R Matrix for id Estimated R Correlation Matrix for id

24 Estimated R Matrix for id Estimated R Correlation Matrix for id Note that in this output, all covariance/correlation matrices are in the order ABC. Comparing this model to the two simpler previous models yields Number of Model B vs. A -2 log L parameters UN separate (0.172) groups UN treatment (0.168) order UN period (0.160) order CS random (0.151) id Testing equality of covariance structures in the three groups (the simpler alternative being either an unstructured covariance for treatments, or an unstructured covariance for periods), we get Treatment-order: 2 log Q = = 9.5 χ 2 (12) P = 0.66 Period-order: 2 log Q = = 14.0 χ 2 (12) P = 0.30 i.e. no indication of a sequence-dependent covariance structure (possibly because the data set is so small). There is even no convincingly argument 24

25 against the simple compound symmetry covariance structure (P=0.10). Comparison of CS to UN-treatment: 2 log Q = = 7.6 χ 2 (4) P = Make an analysis with an appropriate correlation structure, and estimate again the difference in duration between 100 mg and 50 mg tablets, with confidence intervals. If we have to choose between the two sequence-invariant covariance structures, we choose the one with the smallest 2 log L, i.e. the one, where the covariance is determined by the treatment, not the period. We have previously seen the estimate for the difference between 100 mg and 50 mg in this model, namely: (0.168), P= Since all of the covariance structures have agreed upon the significance of this difference, it seems safe to conclude from this investigation regardless of the small sample size. 9. Can we detect any evidence of (one or more) carry-over effects in these data? Since drug C is a sustained-release formulation, we will in advance expect this to show the strongest signs of a carry-over effect, followed by the highdose tablet B. The easy choice to look for a carry-over effect would be just to include an interaction between treatment and period. This is however not very wise: In a two-period cross-over study (with two treatments), this is usually done, but it is important to point out that interaction can arise due to other effects than carry-over. In a three-period design as this, it is however possible to distinguish between the two by defining new variables to model the potential effect of carry over directly. Since we have a three-period design, there may be different carry-over effects depending upon the drug used immediately before, and we therefore define the variable previous1 to hold this information. Note, that the value 25

26 N means that there is no previous treatment. Likewise, we define previous2 to hold information about the treatnent given two periods before. This is of course only different from N for treatments given in period 3. previous1= N ; if group=1 and period=2 then previous1= A ; if group=1 and period=3 then previous1= B ; if group=2 and period=2 then previous1= B ; if group=2 and period=3 then previous1= C ; if group=3 and period=2 then previous1= C ; if group=3 and period=3 then previous1= A ; previous2= N ; if group=1 and period=3 then previous2= A ; if group=2 and period=3 then previous2= B ; if group=3 and period=3 then previous2= C ; With these two variables (previous1 and previous2) as covariates, we must exclude period from the analysis because of confounding, and we therefor make the analysis title specific carry-over ; proc mixed data=a1; class sequence id period treatment previous1 previous2; model duration = treatment previous1 previous2 / outpm=predm s cl; repeated treatment / type=un subject=id r rcorr; yielding the output specific carry-over Model Information Data Set Dependent Variable Covariance Structure Subject Effect Estimation Method Degrees of Freedom Method WORK.A1 duration Unstructured id REML Between-Within 26

27 Class Level Information Class Levels Values sequence 3 ABC BCA CAB id period treatment 3 A B C previous 4 A B C N previous1 4 A B C N previous2 4 A B C N Dimensions Covariance Parameters 6 Columns in X 12 Columns in Z 0 Subjects 12 Max Obs Per Subject 3 Estimated R Matrix for id Estimated R Correlation Matrix for id Solution for Fixed Effects Effect treatment previous1 previous2 Estimate Error DF Intercept treatment A treatment B treatment C 0.. previous1 A previous1 B previous1 C previous1 N 0.. previous2 A previous2 B previous2 C previous2 N 0.. Effect treatment previous1 previous2 t Value Pr > t Alpha Intercept 9.58 < treatment A treatment B treatment C... previous1 A previous1 B previous1 C previous1 N... previous2 A previous2 B previous2 C

28 previous2 N... Effect treatment previous1 previous2 Lower Upper Intercept treatment A treatment B treatment C.. previous1 A previous1 B previous1 C previous1 N.. previous2 A previous2 B previous2 C previous2 N.. Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F treatment previous previous Our predictions from this model is once again identical to the averages, so it is not shown here. We note, that the treatment given in the period immediately before seems to have an effect on the duration on the treatment to follow (P=0.046): Treatments A and C seems to linger on for a while. The treatment given two periods before does not seem to have any influence (P=0.41). If we omit this from the analysis, we have the model: title specific carry-over, only previous1 ; proc mixed data=a1; class sequence id period treatment previous1 previous2; model duration = treatment previous1 / outpm=predm s cl ; repeated treatment / type=un subject=id r rcorr; estimate 100 mg vs 50 mg treatment / cl; which yields the output specific carry-over, only previous1 The Mixed Procedure Class Level Information Class Levels Values 28

29 sequence 3 ABC BCA CAB id period treatment 3 A B C previous1 4 A B C N Dimensions Covariance Parameters 6 Columns in X 8 Columns in Z 0 Subjects 12 Max Obs Per Subject 3 Estimated R Matrix for id Estimated R Correlation Matrix for id Solution for Fixed Effects Effect treatment previous1 Estimate Error DF t Value Intercept treatment A treatment B treatment C 0... previous1 A previous1 B previous1 C previous1 N 0... Effect treatment previous1 Pr > t Alpha Lower Upper Intercept < treatment A treatment B treatment C.... previous1 A previous1 B previous1 C previous1 N.... Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F treatment previous Estimates Label Estimate Error DF t Value Pr > t Alpha 100 mg vs 50 mg

30 Label Lower Upper 100 mg vs 50 mg Based on this output, our conclusion seems to be that treatment C has a carry-over effect, since it prolongs the duration of the treatment to follow by an estimated 0.87 hours, CI=(0.26, 1.49). Our predictions from this reduced model is shown in the figure below: The carry-over effect of A and B are estimated to be quite similar, and not significant, so it is tempting to exclude these (note that we ought to make a formal test for excluding them both simultaneously, a test which would give P=0.11). We do this by defining the new covariate previousc= N ; if group=2 and period=3 then previousc= C ; if group=3 and period=2 then previousc= C ; and using the new model title specific carry-over, only previousc ; proc mixed data=a1; 30

31 class sequence id period treatment previousc previous2; model duration = treatment previousc / outpm=predm s cl ; repeated treatment / type=un subject=id r rcorr; estimate 100 mg vs 50 mg treatment / cl; from which we get the output specific carry-over, only previousc Solution for Fixed Effects previous Effect treatment C Estimate Error DF t Value Pr > t Intercept <.0001 treatment A <.0001 treatment B treatment C previousc C previousc N previous Effect treatment C Alpha Lower Upper Intercept treatment A treatment B treatment C... previousc C previousc N... Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F treatment <.0001 previousc Estimates Label Estimate Error DF t Value Pr > t Alpha 100 mg vs 50 mg Label Lower Upper 100 mg vs 50 mg Based on this, our conclusion is that treatment C definitely shows sign of a carry-over effect, since it prolongs the duration of the treatment to follow by an estimated 0.94 hours, CI=(0.36, 1.51), P= Note also the very important fact, that the estimated effects of the drugs change when we add this carry-over effect to the model! 31