Treatment comparisons in clinical trials with Covariates analysis of diastolic blood pressure

Primary objective of this analysis is to test whether treatment A (new drug) may be effective in lowering diastolic blood pressure (DBP) as compared to treatment B (placebo) and to describe changes in DBP across the times at which it was measured. Diastolic blood pressure data measured in small clinical trials in hypertension, diastolic blood pressure (DBP) was measured (mmHG) in the supine position at baseline (i.e., "DBP1") before randomization and monthly thereafter up to 4 months as indicated by "DBP2,""DBP3,""DBP4" and "DBP5."Patients' age and sex were recorded at baseline and represented as potential covariates.

Figure 1: Boxplot for treatment comparison

In boxplot analysis data appear symmetric and implying that there are no obvious outliers. The new drug treatment A seems to be more effective than the treatment B on average, the DBP decrease for drug A is about 15 mm HG as compared to a decrease of 5 mm HG for drug B. This variation needs to be formally tested to assess whether it is statistically significant or not. This analysis through Statistical significance of t-test (t-test with equal variance).

Two Sampl e t - t est

dat a: di ff by TRTt = - 12. 15, df = 38, p- val ue = 1. 169e- 14al t er nat i ve hypot hesi s: t r ue di ff er ence i n means i s not equal t o 095 per cent confi dence i nt er val : - 12. 132758 - 8. 667242sampl e est i mat es:mean i n gr oup A mean i n gr oup B - 15. 2 - 4. 8

Wel ch Two Sampl e t - t est

dat a: di ff by TRTt = - 12. 15, df = 36. 522, p- val ue = 2. 149e- 14al t er nat i ve hypot hesi s: t r ue di ff er ence i n means i s not equal t o 095 per cent confi dence i nt er val : - 12. 135063 - 8. 664937sampl e est i mat es:mean i n gr oup A mean i n gr oup B - 15. 2 - 4. 8

Table 1: Two sample t test

In above analysis t-statistic is -12.15 with 38 degrees of freedom, p-value is less than 0.01 indicating that the difference is 10.4 (-15.2,-4.8) between the two treatment groups which is strongly statistically significant. We may check the assumption of equal variances through Welch t-test whether the test is significant or not.

Table 2: Welch Two Sample t-test

In Welch two sample t-test analysis, t-statistic is -12.15 with 36.522(the degrees of freedom is no longer an integer and it is close to the degrees of freedom 38 in the previous test) degrees of freedom and p-value is less than 0.01.This suggests that the assumption of equal variances is not really violated for this data set. We may statistically test the null hypothesis of equal variances using the F-test for variances.

F t est t o compar e t wo var i ances

dat a: di ff by TRTF = 1. 5036, num df = 19, denom df = 19, p- val ue = 0. 3819al t er nat i ve hypot hesi s: t r ue r at i o of var i ances i s not equal t o 195 per cent confi dence i nt er val : 0. 595142 3. 798764sampl e est i mat es:r at i o of var i ances

1. 5036

Wi l coxon r ank sum t est wi t h cont i nui t y cor r ect i on

dat a: di ff by TRTW = 0, p- val ue = 6. 286e- 08al t er nat i ve hypot hesi s: t r ue l ocat i on shi f t i s not equal t o 0

Performance of an F test to compare the variances of two samples from normal populations.

Table 3: F-test for variances (null hypothesis of equal variances)

The value of F statistic is 1.50 with degrees of freedom of 19 and p-value is 0.3819. This means that there is insufficient evidence to reject the null hypothesis of equal variances even though the observed variance ratio is 1.50. We check the assumption of equality of variances before performing the standard t-test. If the assumptions of normality and equal variances are violated then we may use the nonparametric version of the t-test (Wilcoxon rank-sum test).

Table 4: Wilcoxon test (Mann-Whitney test) on vectors of the data

In Wilcoxon rank-sum test gives the same conclusions and the one-sided t-test may be more appropriate to test the treatment effect.

One Way ANOVA for time variation In the treatment period and mean changes the months following baseline, we extract the means by treatment group by aggregation.

Wel ch Two Sampl e t - t est

dat a: DBP1 by TRTt = - 0. 30755, df = 37. 829, p- val ue = 0. 7601al t er nat i ve hypot hesi s: t r ue di ff er ence i n means i s not equal t o 095 per cent confi dence i nt er val : - 1. 516666 1. 116666sampl e est i mat es:mean i n gr oup A mean i n gr oup B 116. 55 116. 75

Figure 2: Boxplot for difference in treatment group at DBP1

In diastolic blood pressure (DBP1), age and sex. Plotted at baseline DBP1 for the two treatment groups using a boxplot to see the distributions as well as the difference in treatment g group means. Result shows that a there is no much more difference. Extract the means by treatment group by aggregation through t-test.

Table 5: Welch two sample t-test for the both groups at baselineThe mean DBP at baseline is 116.55 mm HG for treatment group A and 116.75 mm HG for treatment group B. The p-value for the difference (0.20 mm HG) is 0.76, which is not statistically significant. The same test can be performed for Age to conclude the treatment group balance at baseline.

2- sampl e t est f or equal i t y of pr opor t i ons wi t h cont i nui t y cor r ect i on

dat a: SexbyTRTX- squar ed = 0. 10101, df = 1, p- val ue = 0. 7506al t er nat i ve hypot hesi s: t wo. si ded95 per cent confi dence i nt er val : - 0. 4567829 0. 2567829sampl e est i mat es:pr op 1 pr op 2 0. 4 0. 5

Resi dual s: Mi n 1Q Medi an 3Q Max - 2. 1988 - 0. 5601 0. 1693 0. 5245 1. 9506

Coeffi ci ent s: Est i mat e St d. Er r or t val ue Pr ( >| t | ) ( I nt er cept ) 104. 38337 1. 12876 92. 477 < 2e- 16 ***SexM - 0. 64220 0. 30158 - 2. 129 0. 0399 * Age 0. 26388 0. 02284 11. 552 7. 79e- 14 ***- - -Si gni f . codes: 0 ‘ ***’ 0. 001 ‘ **’ 0. 01 ‘ *’ 0. 05 ‘ . ’ 0. 1 ‘ ’ 1

Table 6: 2-sample test for equality of proportions for gender by treatment groupThe p-value for the difference is 0.75 which is not statistically significant for gender by treatment group.

Linear regression for baseline DBP to Age and Sex

Table 6: Linear regression for baseline DBP to Age and Sex

In linear regression analysis age and sex are statistically significant at the 5% level. DBP increases with age for both male and female at the rate of 0.264 mm HG.

In this plot two levels represent the sex and the solid line is the regression line for female, the dashed line is the regression line for male, the DBP increases with age for the both gender.

Figure 3: Regression plotting the DBP1 with age as covariate

Regression model with the all covariates

Table 7: Linear regression model with all covariates

Table 8: Stepwise linear regression model

Stepwise model selection, these two 2-way interactions are not statistically significant.

Resi dual s: Mi n 1Q Medi an 3Q Max - 5. 5636 - 1. 2965 - 0. 0199 1. 2272 5. 3584

Coeffi ci ent s: Est i mat e St d. Er r or t val ue Pr ( >| t | )( I nt er cept ) - 5. 0883 6. 6441 - 0. 766 0. 449TRTB - 0. 8316 8. 4748 - 0. 098 0. 922Age - 0. 2156 0. 1348 - 1. 600 0. 119SexM 5. 5952 9. 3802 0. 596 0. 555TRTB: Age 0. 2347 0. 1728 1. 358 0. 184TRTB: SexM 7. 1304 11. 9501 0. 597 0. 555Age: SexM - 0. 1030 0. 1914 - 0. 538 0. 594TRTB: Age: SexM - 0. 1641 0. 2467 - 0. 665 0. 511

Resi dual st andar d er r or : 2. 473 on 32 degr ees of f r eedomMul t i pl e R- squar ed: 0. 8561, Adj ust ed R- squar ed: 0. 8246 F- st at i st i c: 27. 19 on 7 and 32 DF, p- val ue: 9. 411e- 12

St ar t : AI C=79. 52di ff ~ TRT * Age * Sex Df Sum of Sq RSS AI C- TRT: Age: Sex 1 2. 7059 198. 47 78. 07<none> 195. 76 79. 52St ep: AI C=78. 07di ff ~ TRT + Age + Sex + TRT: Age + TRT: Sex + Age: Sex Df Sum of Sq RSS AI C- TRT: Sex 1 1. 3256 199. 79 76. 336- TRT: Age 1 9. 5638 208. 03 77. 952<none> 198. 47 78. 070- Age: Sex 1 17. 0694 215. 53 79. 370

St ep: AI C=76. 34

In the fitted reduced model analysis, we see that TRT and Age are statistically significant. The model p-value is 4.24e-15 with R2 = 0.833 indicating satisfactory model and significant.

Table 9: Fitted reduced model (ANOVA Table)

Table 10: Fitted reduced model

Anal ysi s of Var i ance Tabl e

Response: di ff Df Sum Sq Mean Sq F val ue Pr ( >F) TRT 1 1081. 60 1081. 60 176. 0395 1. 228e- 15 ***Age 1 51. 07 51. 07 8. 3119 0. 006525 ** Resi dual s 37 227. 33 6. 14 - - -Si gni f . codes: 0 ‘ ***’ 0. 001 ‘ **’ 0. 01 ‘ *’ 0. 05 ‘ . ’ 0. 1 ‘ ’ 1

Resi dual s: Mi n 1Q Medi an 3Q Max - 5. 9039 - 1. 6516 - 0. 0091 1. 1557 5. 2299

Coeffi ci ent s: Est i mat e St d. Er r or t val ue Pr ( >| t | ) ( I nt er cept ) - 6. 78086 2. 97236 - 2. 281 0. 02838 * TRTB 10. 13149 0. 78936 12. 835 3. 38e- 15 ***Age - 0. 17323 0. 06009 - 2. 883 0. 00653 ** - - -Si gni f . codes: 0 ‘ ***’ 0. 001 ‘ **’ 0. 01 ‘ *’ 0. 05 ‘ . ’ 0. 1 ‘ ’ 1

Resi dual st andar d er r or : 2. 479 on 37 degr ees of f r eedomMul t i pl e R- squar ed: 0. 8328, Adj ust ed R- squar ed: 0. 8238 F- st at i st i c: 92. 18 on 2 and 37 DF, p- val ue: 4. 243e- 15

In fitted reduced model plotting, the new treatment (A) dropped DBP by 10 mm HG compared to placebo (B). Age is a significant covariate and shows a DBP decreasing trend.

Figure 4: Plotted fitted reduced model for both treatment Age is a significant covariate and shows a DBP decreasing trend