ABBREVIATIONS:

A Active Drug

B Placebo

Baseline BP B0

BP Blood Pressure [in mmHG]

CV Cardio Vascular Disease

DOD Difference of Difference (T test)

LM linear model

TRT Treatment either placebo or Drug Beta blocker

RCT Randomized Clinical Trial

RM Repeated Measures

RTM Regression to the Mean

SD Standard deviation

GOAL

The study of longitudinal data is a challenge for Scientists and practioner beacause one want to make inference as t o account for RM : Classic statistical tests withjout accounting for it is a flaw in publications and often seen.

The main subject here is how including a baseline as a covariate act on your model prediction.

We present a short analysis as follow:

RCT & Data. Wide vs Long format
Grahs ID trajectories or Growth curve
T Test on endpoint 4 months
Some word on variability in time
Linear model (with/without baseline inclusion)
Linear mixed model

Rct and DATA

Beta blocker and Reserpine like drugs¹ has been proven significantly efficient in CV diseases (Stroke,Infarction,BP reduction etc), this since the 60’s which was the golden age of these discoveries.

The physiological mechanisms are:

B1 or VMAST calcic antagonist blocking agent² reduce contractile ventricular pumping hence reducing the load on arterial and peripheral pipe systems
Less blood flow
Inhibition of renine-angiotensine II at kidney receptor site -An reduce autonmous nervous system activity.
Vasodilatation of vessels and much more effects

Of course multi targeted spots in our cells receptors (via Calcium or B blockade chained) result in ends of a major drop in our BP.

The data presented here come from a study: Veterans Administration Cooperative Study Group on Anti hypertensive Agents (VA Study Group (1967, D. Freis):
Endpoints Y_it: Diastolic BP Supine horizontal in [mmHg]

Beside our BP follwows cycles and might be influenced by: Natural Rhythm (“random”) , Emotions-Stress (“random”) , Cafeine-Alcholl-drugs-exercises consumption (can be controlled) among others.
Randomization between placebo and TRT were done.
Questions or interest : Does the new Drug reduce the Diastolic BP with time?

Covariate : sex,age and a baseline [BP0] record.

Response: Y_it are the BP at 1 months intervals: BP0 BP1 BP2 BP3 BP4

Duration 4 months follow-up from Baseline inwards.

Note:Data from ID1 was removed to create unbalanced design and show up difficulties in running within subject ANOVA..

Note that Baseline is one a covariate once as a real measure in time=0 and we will explore this possibilities for inefrences and modelling.

Know your Data: Systematic review

The data come form the book CLINICAL DATA ANALYSIS USING R (Chen & Peace , 2011) data extracted from the 1967 Trial: Some question arouse:
Does the author have copied partially the data? Does it exist an metadat of teh DOE?

Such questions must be answered before procedinngs to the analysis.

Note that 20 id in Placebo group B and 20[-1]³ id in trt group [A] seems to be a phase II trial as n is low and the reader should bear in mind at this stage of analysis non significant result might come from the too low power of the study design, especially in longitudinal data.

In present study reader must be aware that regression to the mean might occur when :

1: When part of a population with abnormal value are selected

2: andom process occur under Normal law

3: In repeated measures RTM is exclusively noticeable

Note : could be attributed to the Placebo effect from practitioner

By RCT design we expect that RTM is the same magnitude in both arms and hence is accounted for in assignment of trt.

From the author:

We wont account for sake of simplicity in the design, study of within between subject and LMM will be exposed in other blog Using lm in Longitudinal data is a flaw as RM are correlated but coefficients are ussually usable but the SE for inferences.

We also try to use different plotting packages i.e ggplot2 and lattice to present the reader the pro and cons of each one.(let at your appraisal)

Data summary

## [1] "GROUP A= TRTEATMENT B= PLACEBO"

RAW DATA BP
id	trt	baseline	bp1	bp2	bp3	bp4	age	sex
2	A	116	113	112	103	101	51	M
3	A	119	115	113	104	98	48	F
4	A	115	113	112	109	101	42	F
5	A	116	112	107	104	105	49	M
6	A	117	112	113	104	102	47	M
7	A	118	111	100	109	99	50	F
8	A	120	115	113	102	102	61	M
9	A	114	112	113	109	103	43	M
10	A	115	113	108	106	97	51	M
11	A	117	112	110	109	101	47	F
12	A	116	115	113	109	102	45	M
13	A	119	117	110	106	104	54	F
14	A	118	115	113	102	99	52	M
15	A	115	112	108	105	102	42	M
16	A	114	111	111	107	100	44	F
17	A	117	114	110	108	102	48	M
18	A	120	115	113	107	103	63	F
19	A	114	113	109	104	100	41	M
20	A	117	115	113	109	101	51	M
21	B	114	115	113	111	113	39	M
22	B	116	114	114	109	110	40	F
23	B	114	115	113	111	109	39	F
24	B	114	115	113	114	115	38	M
25	B	116	113	113	109	109	39	F
26	B	114	115	114	111	110	41	M
27	B	119	118	118	117	115	56	F
28	B	118	117	117	116	112	56	M
29	B	114	113	113	109	108	38	M
30	B	120	115	113	113	113	57	M
31	B	117	115	113	114	115	47	F
32	B	118	114	112	109	110	48	M
33	B	121	119	117	114	115	61	F
34	B	116	115	116	114	111	49	M
35	B	118	118	113	113	112	52	M
36	B	119	115	115	114	111	55	F
37	B	116	114	113	109	109	45	F
38	B	116	115	114	114	112	42	M
39	B	117	115	113	114	115	49	F
40	B	118	114	114	114	115	50	F

SUMMARY WIDE DATA BP.CSV
id	trt	baseline	bp1	bp2	bp3	bp4	age	sex
Min. : 2.0	A:19	Min. :114.0	Min. :111.0	Min. :100.0	Min. :102.0	Min. : 97.0	Min. :38.00	Length:39
1st Qu.:11.5	B:20	1st Qu.:115.0	1st Qu.:113.0	1st Qu.:112.0	1st Qu.:106.5	1st Qu.:101.5	1st Qu.:42.00	Class :character
Median :21.0	NA	Median :117.0	Median :115.0	Median :113.0	Median :109.0	Median :108.0	Median :48.00	Mode :character
Mean :21.0	NA	Mean :116.7	Mean :114.3	Mean :112.4	Mean :109.4	Mean :106.7	Mean :47.95	NA
3rd Qu.:30.5	NA	3rd Qu.:118.0	3rd Qu.:115.0	3rd Qu.:113.0	3rd Qu.:113.5	3rd Qu.:112.0	3rd Qu.:51.50	NA
Max. :40.0	NA	Max. :121.0	Max. :119.0	Max. :118.0	Max. :117.0	Max. :115.0	Max. :63.00	NA

## [1] "How to process data from wide to long format creating a time variable in one RM variable is given in the attached code"

Baseline B0 value

Baseline B0 value is about the same for the two groups that is SRS Randomization on baseline value is correct, and matching of covariates seems reasonable (Age ,Sex)
Baseline value are slightly higher for older as evidence base physiology (Rigidity artery aging i-e.)
BP increased with aging is proven but in this study is disregarded for study under 4 months time: Hence Age will be a fixed covariates in time.
Baseline value are slightly higher range for women in Placebo.

Contrasts:

To have a possible lowering trend value of the active trt contrast should be done as “TREATMENT” contrast to the reference Placebo group = B as reference level factor.We expect under this kind of contrast a negative regression coefficient for the active drug A if effective.

Remark:

If you want an orthogonal testing lm ANOVA type III, contrast must be set to sum!.

## [1] "Range baseline A"

## [1] 114 120

## [1] 114 121

##   B
## A 0
## B 1

##   A
## B 0
## A 1

Both trt result in lowering BP after 4M The magnitude is greater in trt A Drug group The is more linearity in that group , one erratic pattern In the A group e step down slop occurs after 2 months of active drug Conversely flattening occurs in the Placebo group after 2 months. A linear assumptions might be a crude assumptions but for a accurate modelling ; piece linear of polynomial trend should be envisioned to capture the pattern at 2-3 months periods.

Within and Between variability

about sd within and between sphericity

Mean.TRT	Mean.baseline	Mean.bp1	Mean.bp2	Mean.bp3	Mean.bp4	sd.TRT	sd.baseline	sd.bp1	sd.bp2	sd.bp3	sd.bp4
A	116.6842	113.4211	110.5789	106.1053	101.1579	A	1.945154	1.677160	3.271354	2.558097	2.007297
B	116.7500	115.2000	114.0500	112.4500	111.9500	B	2.124419	1.609184	1.669384	2.502104	2.438183

## [1] 0.534202

## 
##  Bartlett test of homogeneity of variances
## 
## data:  BP[, c(3, 4, 5, 6, 7)]
## Bartlett's K-squared = 68.817, df = 4, p-value = 4.033e-14

Analysis :From Baseline to a 4 Months time point [Naive⁴]

Effect of Drug in lowering is BP at 4 months time is obvious in graphs:
Dotplot + Boxplot show clear clustered drops - no overlap - no outlier
Placebo[B] show no 4th quartile
Slope of DRUG A are steeper and more parallel
Placebo exhibit lower value drop and greater variability in slope :
Note: Keep scaling and range as fixed for both graph Note that on Placebo group several value of Baseline are duplicated despite aspect of that graph check it with command (duplicated(BP$baseline and dp4[BP$trt==“B”]))

Some comparison at Baseline difference:

Looking for the effect

## [1] "BP baseline value"

##        A        B 
## 116.6842 116.7500

## [1] "sd B0 value"

##        A        B 
## 2.007297 2.438183

## [1] "Difference from baseline to 4M"

##        A        B 
## 15.56211  4.77000

## [1] "A drop of 15 mmHG might be expected by trt A vs 5 mmHG"

Baseline value mean: around 116.7 both arms: B 116.75 / A 116.68 mmHG: We dont know if pairing occured

Drop from baseline mean at 4M trt mean: B 4.77 / A 15.56 mmHG

Pooled SD about: 2.2 mmHG

That is:

Drop is greater than resp.pooled sd
High effect size might be expected even in PLA group.
Design experiment plans usually a 10 mmHg drop as first clinical intention(i.e Captopril ECA inhibitor).
Note:

Usually we expect in each measurements a great variability/instability of BP monthly measure (protocol, devices used)…Here the respective sd for the time variable (each months) is pretty stable and low (less than 5mmHG): That might be a signed that the mean of several BP at each visit were made (?).Reseracher has to go back in the design of the experiment to confirmed it and have a physiological knowledge of the involved process:

A B

0 1.945154 2.124419

1 1.677160 1.609184

2 3.271354 1.669384

3 2.558097 2.502104

4 2.007297 2.438183

	A	B
0	1.945154	2.124419
1	1.677160	1.609184
2	3.271354	1.669384
3	2.558097	2.502104
4	2.007297	2.438183

Inference: T.Test

Although not planned to discuss deeply in this blog we report two type of t.test:

2 sample T test pooled variance at 4M [_Naive _approach]
DoD T.test pooled variance

Lets graphs first plot the difference time M to baseline value:

The interpretation is left at the reader : However note that usually we expect with DOD more precise and difference in the result but here not as much as expected.

Looking at correlation / covariance in both arms presented below and which value is below <0.5 might explain why such result occurs. Differentiating is not removing as much covariance from the DoD test VAR[X-Y] = VARX + VARY - 2 COV XY

This might be a sign also to researcher, despite parallel plot pattern a linear assumptions between baseline and time point 4 might be doubtful due to low correlations (See xyplot by id below).

## 
##  Two Sample t-test
## 
## data:  BP$bp4 by BP$trt
## t = -15.046, df = 37, p-value < 2.2e-16
## alternative hypothesis: true difference in means between group A and group B is not equal to 0
## 95 percent confidence interval:
##  -12.245436  -9.338774
## sample estimates:
## mean in group A mean in group B 
##        101.1579        111.9500

## 
##  Two Sample t-test
## 
## data:  BP$baseline - BP$bp4 by BP$trt
## t = 13.203, df = 37, p-value = 1.427e-15
## alternative hypothesis: true difference in means between group A and group B is not equal to 0
## 95 percent confidence interval:
##   9.080235 12.372397
## sample estimates:
## mean in group A mean in group B 
##        15.52632         4.80000

True difference in trt time points 4 two sample T test :

[9.34-12.24]

True difference in DOD T test:

[9.08-12.37]

Correlations

## [1] 0.09885125

## [1] 0.4445472

## [1] 0.127816

TRT	COR B0-BP4M
A	0.10
B	0.45
OVERALL	0.12

Linear model with wave time points⁵

Response feature analysis is let at the reader options

Let first explore the possibility of an interactions time * trt effect: As mentioned Interactions that is slope in time with trt can be visualized as suchon the interaction plot.

Therefore it might be assessed that trt difference between arms is effective in time elapsed: (fan out sloping)

## # A tibble: 10 × 3
## # Groups:   time [5]
##     time trt   tip_groups
##    <dbl> <fct>      <dbl>
##  1     0 A           117.
##  2     0 B           117.
##  3     1 A           113.
##  4     1 B           115.
##  5     2 A           111.
##  6     2 B           114.
##  7     3 A           106.
##  8     3 B           112.
##  9     4 A           101.
## 10     4 B           112.

LINEAR MODEL TRT VS TRT BASELINE

library(kableExtra)
summary(lm(BP$bp4~BP$trt))

## 
## Call:
## lm(formula = BP$bp4 ~ BP$trt)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -4.158 -1.950  0.050  1.446  3.842 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 101.1579     0.5136  196.94   <2e-16 ***
## BP$trtB      10.7921     0.7173   15.05   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.239 on 37 degrees of freedom
## Multiple R-squared:  0.8595, Adjusted R-squared:  0.8557 
## F-statistic: 226.4 on 1 and 37 DF,  p-value: < 2.2e-16

summary(lm(BP$bp4~BP$trt+BP$baseline))#controlling for baseline

## 
## Call:
## lm(formula = BP$bp4 ~ BP$trt + BP$baseline)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.9210 -1.6971 -0.2505  1.5233  4.0676 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  62.7089    20.3743   3.078  0.00397 ** 
## BP$trtB      10.7704     0.6937  15.525  < 2e-16 ***
## BP$baseline   0.3295     0.1746   1.888  0.06715 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.165 on 36 degrees of freedom
## Multiple R-squared:  0.8722, Adjusted R-squared:  0.8651 
## F-statistic: 122.8 on 2 and 36 DF,  p-value: < 2.2e-16

m1=lm(BP$bp4~BP$trt)
m2=lm(BP$bp4~BP$trt+BP$baseline)
equatiomatic::extract_eq(m1,use_coefs = TRUE)

\[ \operatorname{\widehat{BP\$bp4}} = 101.16 + 10.79() \]

equatiomatic::extract_eq(m2,use_coefs = TRUE)

\[ \operatorname{\widehat{BP\$bp4}} = 62.71 + 10.77() + 0.33(\operatorname{BP\$baseline}) \]

round(confint(m1),2)

##              2.5 % 97.5 %
## (Intercept) 100.12 102.20
## BP$trtB       9.34  12.25

round(confint(m2),2)

##             2.5 % 97.5 %
## (Intercept) 21.39 104.03
## BP$trtB      9.36  12.18
## BP$baseline -0.02   0.68

Fitted value

Anova model comparison

## Anova Table (Type III tests)
## 
## Response: BP$bp4
##              Sum Sq Df  F value    Pr(>F)    
## (Intercept)   44.41  1   9.4731  0.003973 ** 
## BP$baseline   16.71  1   3.5634  0.067149 .  
## BP$trt      1129.97  1 241.0299 < 2.2e-16 ***
## Residuals    168.77 36                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

##             Df Sum Sq Mean Sq F value Pr(>F)    
## BP$trt       1 1134.8  1134.8 242.067 <2e-16 ***
## BP$baseline  1   16.7    16.7   3.563 0.0671 .  
## Residuals   36  168.8     4.7                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

TRT is highly significant in both models.

On simple linar model with TRT is a factor the beta coefficients reflect the decrease in mean in BP from placebo (contr trt ref PLA) There is some gain including the baseline m2 in the coefficient and in SE.

Baseline will be considered as a Fixed control covariate, not a time covariate.

Note An Interaction as no meaning in baseline and trt due to randomization and equal value.

Note the T value for baseline is the same for anova model comparison

Note the Anova model comparison is a limited value here anova(m1,m2)

A little gain is done here in this dataset with but with prediction points a lot is done by adding a continuous covariates (ANCOVA s.lato) vs an ANOVA lm as E(Y|X) is always better that the mean.

However the prediction of the drug A is fairly in accordance with true value.

We saw a slight but negligible difference in type I and III due to missing ID1 unbalanced.

Now lets explore an DOD Y|X = (BP4 BPbaseline) “like” regression with and without baseline in the X covariate

LINEAR MODEL with Y as Differnece from baseline

With-without baseline inclusion on covariate Xi also!

## 
## Call:
## lm(formula = BP$diff4b ~ BP$trt)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -5.474 -1.837 -0.200  1.526  5.800 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -15.5263     0.5818  -26.69  < 2e-16 ***
## BP$trtB      10.7263     0.8124   13.20 1.43e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.536 on 37 degrees of freedom
## Multiple R-squared:  0.8249, Adjusted R-squared:  0.8202 
## F-statistic: 174.3 on 1 and 37 DF,  p-value: 1.427e-15

\[ \operatorname{\widehat{BP\$bp4}} = 101.16 + 10.79() \] \[ \operatorname{\widehat{BP\$bp4}} = 62.71 + 10.77() + 0.33(\operatorname{BP\$baseline}) \]

## 
## Call:
## lm(formula = BP$diff4b ~ BP$baseline + BP$trt)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.9210 -1.6971 -0.2505  1.5233  4.0676 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  62.7089    20.3743   3.078 0.003973 ** 
## BP$baseline  -0.6705     0.1746  -3.841 0.000478 ***
## BP$trtB      10.7704     0.6937  15.525  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.165 on 36 degrees of freedom
## Multiple R-squared:  0.8758, Adjusted R-squared:  0.8689 
## F-statistic: 126.9 on 2 and 36 DF,  p-value: < 2.2e-16

##              2.5 % 97.5 %
## (Intercept) -16.71 -14.35
## BP$trtB       9.08  12.37

##             2.5 % 97.5 %
## (Intercept) 21.39 104.03
## BP$baseline -1.02  -0.32
## BP$trtB      9.36  12.18

Fitted value

LONGITUDINAL DATA ANALYSIS

Adding a static baseline in long format

## 
## Call:
## lm(formula = Score ~ time, data = new)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.8923  -2.3897   0.1077   2.1051   8.1128 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 116.8974     0.4560  256.37   <2e-16 ***
## time         -2.5026     0.1861  -13.44   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.676 on 193 degrees of freedom
## Multiple R-squared:  0.4836, Adjusted R-squared:  0.4809 
## F-statistic: 180.7 on 1 and 193 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = Score ~ time + trt, data = new)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.5895 -1.8347 -0.0851  2.4054  5.9251 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 114.5946     0.4195  273.16   <2e-16 ***
## time         -2.5026     0.1474  -16.98   <2e-16 ***
## trtB          4.4905     0.4169   10.77   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.91 on 192 degrees of freedom
## Multiple R-squared:  0.6781, Adjusted R-squared:  0.6747 
## F-statistic: 202.2 on 2 and 192 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = Score ~ time * trt, data = new)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.5895 -1.4263 -0.2632  1.5737  4.4500 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 117.2632     0.3997 293.383   <2e-16 ***
## time         -3.8368     0.1632 -23.514   <2e-16 ***
## trtB         -0.7132     0.5581  -1.278    0.203    
## time:trtB     2.6018     0.2279  11.419   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.249 on 191 degrees of freedom
## Multiple R-squared:  0.8087, Adjusted R-squared:  0.8057 
## F-statistic: 269.1 on 3 and 191 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = Score ~ poly(time, 3), data = new)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -12.119  -2.493   0.322   2.101   8.348 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    111.8923     0.2643 423.399   <2e-16 ***
## poly(time, 3)1 -49.4217     3.6904 -13.392   <2e-16 ***
## poly(time, 3)2  -2.6534     3.6904  -0.719    0.473    
## poly(time, 3)3  -0.1519     3.6904  -0.041    0.967    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.69 on 191 degrees of freedom
## Multiple R-squared:  0.485,  Adjusted R-squared:  0.4769 
## F-statistic: 59.96 on 3 and 191 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = Score ~ poly(time, 2) * trt, data = new)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.3038  -1.5489  -0.1971   1.2729   4.2729 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)         109.5895     0.2274 481.934  < 2e-16 ***
## poly(time, 2)1      -75.7716     3.1754 -23.862  < 2e-16 ***
## poly(time, 2)2       -8.3452     3.1754  -2.628  0.00929 ** 
## trtB                  4.4905     0.3175  14.142  < 2e-16 ***
## poly(time, 2)1:trtB  51.3823     4.4342  11.588  < 2e-16 ***
## poly(time, 2)2:trtB  11.0992     4.4342   2.503  0.01316 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.216 on 189 degrees of freedom
## Multiple R-squared:  0.8162, Adjusted R-squared:  0.8113 
## F-statistic: 167.8 on 5 and 189 DF,  p-value: < 2.2e-16

## Analysis of Variance Table
## 
## Model 1: Score ~ poly(time, 2) * trt
## Model 2: Score ~ poly(time, 3)
##   Res.Df     RSS Df Sum of Sq      F    Pr(>F)    
## 1    189  928.43                                  
## 2    191 2601.17 -2   -1672.7 170.26 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

## Linear mixed-effects model fit by REML
##   Data: new 
##        AIC      BIC    logLik
##   863.3417 882.8554 -425.6709
## 
## Random effects:
##  Formula: ~1 | id
##         (Intercept) Residual
## StdDev:    1.178514 1.927075
## 
## Fixed effects:  Score ~ time * trt 
##                 Value Std.Error  DF   t-value p-value
## (Intercept) 117.26316 0.4363163 154 268.75724  0.0000
## time         -3.83684 0.1398047 154 -27.44430  0.0000
## trtB         -0.71316 0.6092825  37  -1.17049  0.2493
## time:trtB     2.60184 0.1952266 154  13.32729  0.0000
##  Correlation: 
##           (Intr) time   trtB  
## time      -0.641              
## trtB      -0.716  0.459       
## time:trtB  0.459 -0.716 -0.641
## 
## Standardized Within-Group Residuals:
##         Min          Q1         Med          Q3         Max 
## -4.23589250 -0.55319361 -0.04672961  0.53978532  2.42541707 
## 
## Number of Observations: 195
## Number of Groups: 39

## Linear mixed-effects model fit by REML
##   Data: new 
##        AIC      BIC    logLik
##   866.1673 892.1855 -425.0837
## 
## Random effects:
##  Formula: ~time | id
##  Structure: General positive-definite, Log-Cholesky parametrization
##             StdDev   Corr  
## (Intercept) 1.113436 (Intr)
## time        0.276513 -0.097
## Residual    1.878806       
## 
## Fixed effects:  Score ~ time * trt 
##                 Value Std.Error  DF   t-value p-value
## (Intercept) 117.26316 0.4203812 154 278.94482  0.0000
## time         -3.83684 0.1503418 154 -25.52079  0.0000
## trtB         -0.71316 0.5870304  37  -1.21486  0.2321
## time:trtB     2.60184 0.2099409 154  12.39321  0.0000
##  Correlation: 
##           (Intr) time   trtB  
## time      -0.613              
## trtB      -0.716  0.439       
## time:trtB  0.439 -0.716 -0.613
## 
## Standardized Within-Group Residuals:
##         Min          Q1         Med          Q3         Max 
## -4.32273721 -0.56636844 -0.04703011  0.52025097  2.58014393 
## 
## Number of Observations: 195
## Number of Groups: 39

It is questionable in lmer model that dual ,interactions as fixed effects and RIAS slope model might be overfitting. Well the options here is as the grapgh show obvisouly andfixed effect interaction (interaction plot) but the respective id curve shows no obvious erratic change in slope population RIAS might be diseragred for simplicity (RAZOR AKHAM).

Now including a baseline as fixed covariate (long format R)

Wrong way to do it: Wide and bp as factor time!

## 
## Call:
## lm(formula = BP$bp4 ~ BP$bp1 + BP$bp3 + BP$baseline + BP$trt)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.7508 -1.3071 -0.3603  1.4835  5.0010 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  19.9953    24.9821   0.800    0.429    
## BP$bp1        0.2199     0.2498   0.881    0.385    
## BP$bp3        0.3484     0.1328   2.623    0.013 *  
## BP$baseline   0.1650     0.1971   0.837    0.409    
## BP$trtB       8.1794     1.0747   7.611 7.64e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.983 on 34 degrees of freedom
## Multiple R-squared:  0.8988, Adjusted R-squared:  0.8869 
## F-statistic: 75.47 on 4 and 34 DF,  p-value: < 2.2e-16

One need to cast the data into long format , required format for longitudinal data:

##  [1] 116 119 115 116 117 118 120 114 115 117 116 119 118 115 114 117 120 114 117
## [20] 114 116 114 114 116 114 119 118 114 120 117 118 121 116 118 119 116 116 117
## [39] 118

##   [1] 116 119 115 116 117 118 120 114 115 117 116 119 118 115 114 117 120 114
##  [19] 117 114 116 114 114 116 114 119 118 114 120 117 118 121 116 118 119 116
##  [37] 116 117 118 116 119 115 116 117 118 120 114 115 117 116 119 118 115 114
##  [55] 117 120 114 117 114 116 114 114 116 114 119 118 114 120 117 118 121 116
##  [73] 118 119 116 116 117 118 116 119 115 116 117 118 120 114 115 117 116 119
##  [91] 118 115 114 117 120 114 117 114 116 114 114 116 114 119 118 114 120 117
## [109] 118 121 116 118 119 116 116 117 118 116 119 115 116 117 118 120 114 115
## [127] 117 116 119 118 115 114 117 120 114 117 114 116 114 114 116 114 119 118
## [145] 114 120 117 118 121 116 118 119 116 116 117 118 116 119 115 116 117 118
## [163] 120 114 115 117 116 119 118 115 114 117 120 114 117 114 116 114 114 116
## [181] 114 119 118 114 120 117 118 121 116 118 119 116 116 117 118

## 
## Call:
## lm(formula = Score ~ time, data = new)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.8923  -2.3897   0.1077   2.1051   8.1128 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 116.8974     0.4560  256.37   <2e-16 ***
## time         -2.5026     0.1861  -13.44   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.676 on 193 degrees of freedom
## Multiple R-squared:  0.4836, Adjusted R-squared:  0.4809 
## F-statistic: 180.7 on 1 and 193 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = Score ~ time + trt, data = new)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.5895 -1.8347 -0.0851  2.4054  5.9251 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 114.5946     0.4195  273.16   <2e-16 ***
## time         -2.5026     0.1474  -16.98   <2e-16 ***
## trtB          4.4905     0.4169   10.77   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.91 on 192 degrees of freedom
## Multiple R-squared:  0.6781, Adjusted R-squared:  0.6747 
## F-statistic: 202.2 on 2 and 192 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = Score ~ time + trt + b0, data = new)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.1859  -2.1675   0.2599   2.1481   7.1716 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 61.70601   11.66924   5.288 3.37e-07 ***
## time        -2.50256    0.14037 -17.828  < 2e-16 ***
## trtB         4.46071    0.39722  11.230  < 2e-16 ***
## b0           0.45326    0.09995   4.535 1.02e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.772 on 191 degrees of freedom
## Multiple R-squared:  0.7094, Adjusted R-squared:  0.7048 
## F-statistic: 155.4 on 3 and 191 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = Score ~ time * trt, data = new)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.5895 -1.4263 -0.2632  1.5737  4.4500 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 117.2632     0.3997 293.383   <2e-16 ***
## time         -3.8368     0.1632 -23.514   <2e-16 ***
## trtB         -0.7132     0.5581  -1.278    0.203    
## time:trtB     2.6018     0.2279  11.419   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.249 on 191 degrees of freedom
## Multiple R-squared:  0.8087, Adjusted R-squared:  0.8057 
## F-statistic: 269.1 on 3 and 191 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = Score ~ time * trt + b0, data = new)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.1859  -1.2015  -0.0998   1.1424   4.6365 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 64.37457    8.68442   7.413 3.97e-12 ***
## time        -3.83684    0.14963 -25.643  < 2e-16 ***
## trtB        -0.74298    0.51182  -1.452    0.148    
## b0           0.45326    0.07436   6.095 5.96e-09 ***
## time:trtB    2.60184    0.20894  12.453  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.062 on 190 degrees of freedom
## Multiple R-squared:   0.84,  Adjusted R-squared:  0.8366 
## F-statistic: 249.3 on 4 and 190 DF,  p-value: < 2.2e-16

## Analysis of Variance Table
## 
## Model 1: Score ~ time * trt
## Model 2: Score ~ time * trt + b0
##   Res.Df    RSS Df Sum of Sq      F   Pr(>F)    
## 1    191 966.25                                 
## 2    190 808.20  1    158.05 37.155 5.96e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

##                  2.5 %      97.5 %
## (Intercept) 116.474779 118.0515364
## time         -4.158696  -3.5149879
## trtB         -1.814069   0.3877528
## time:trtB     2.152397   3.0512870

##                  2.5 %     97.5 %
## (Intercept) 47.2442980 81.5048321
## time        -4.1319830 -3.5417013
## trtB        -1.7525610  0.2666053
## b0           0.3065846  0.5999406
## time:trtB    2.1897003  3.0139839

## 
## Call:
## lm(formula = Score ~ poly(time, 3), data = new)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -12.119  -2.493   0.322   2.101   8.348 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    111.8923     0.2643 423.399   <2e-16 ***
## poly(time, 3)1 -49.4217     3.6904 -13.392   <2e-16 ***
## poly(time, 3)2  -2.6534     3.6904  -0.719    0.473    
## poly(time, 3)3  -0.1519     3.6904  -0.041    0.967    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.69 on 191 degrees of freedom
## Multiple R-squared:  0.485,  Adjusted R-squared:  0.4769 
## F-statistic: 59.96 on 3 and 191 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = Score ~ poly(time, 2) * trt, data = new)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.3038  -1.5489  -0.1971   1.2729   4.2729 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)         109.5895     0.2274 481.934  < 2e-16 ***
## poly(time, 2)1      -75.7716     3.1754 -23.862  < 2e-16 ***
## poly(time, 2)2       -8.3452     3.1754  -2.628  0.00929 ** 
## trtB                  4.4905     0.3175  14.142  < 2e-16 ***
## poly(time, 2)1:trtB  51.3823     4.4342  11.588  < 2e-16 ***
## poly(time, 2)2:trtB  11.0992     4.4342   2.503  0.01316 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.216 on 189 degrees of freedom
## Multiple R-squared:  0.8162, Adjusted R-squared:  0.8113 
## F-statistic: 167.8 on 5 and 189 DF,  p-value: < 2.2e-16

## Analysis of Variance Table
## 
## Model 1: Score ~ poly(time, 2) * trt
## Model 2: Score ~ poly(time, 3)
##   Res.Df     RSS Df Sum of Sq      F    Pr(>F)    
## 1    189  928.43                                  
## 2    191 2601.17 -2   -1672.7 170.26 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

## Linear mixed-effects model fit by REML
##   Data: new 
##        AIC      BIC    logLik
##   863.3417 882.8554 -425.6709
## 
## Random effects:
##  Formula: ~1 | id
##         (Intercept) Residual
## StdDev:    1.178514 1.927075
## 
## Fixed effects:  Score ~ time * trt 
##                 Value Std.Error  DF   t-value p-value
## (Intercept) 117.26316 0.4363163 154 268.75724  0.0000
## time         -3.83684 0.1398047 154 -27.44430  0.0000
## trtB         -0.71316 0.6092825  37  -1.17049  0.2493
## time:trtB     2.60184 0.1952266 154  13.32729  0.0000
##  Correlation: 
##           (Intr) time   trtB  
## time      -0.641              
## trtB      -0.716  0.459       
## time:trtB  0.459 -0.716 -0.641
## 
## Standardized Within-Group Residuals:
##         Min          Q1         Med          Q3         Max 
## -4.23589250 -0.55319361 -0.04672961  0.53978532  2.42541707 
## 
## Number of Observations: 195
## Number of Groups: 39

## Linear mixed-effects model fit by REML
##   Data: new 
##        AIC      BIC    logLik
##   866.1673 892.1855 -425.0837
## 
## Random effects:
##  Formula: ~time | id
##  Structure: General positive-definite, Log-Cholesky parametrization
##             StdDev   Corr  
## (Intercept) 1.113436 (Intr)
## time        0.276513 -0.097
## Residual    1.878806       
## 
## Fixed effects:  Score ~ time * trt 
##                 Value Std.Error  DF   t-value p-value
## (Intercept) 117.26316 0.4203812 154 278.94482  0.0000
## time         -3.83684 0.1503418 154 -25.52079  0.0000
## trtB         -0.71316 0.5870304  37  -1.21486  0.2321
## time:trtB     2.60184 0.2099409 154  12.39321  0.0000
##  Correlation: 
##           (Intr) time   trtB  
## time      -0.613              
## trtB      -0.716  0.439       
## time:trtB  0.439 -0.716 -0.613
## 
## Standardized Within-Group Residuals:
##         Min          Q1         Med          Q3         Max 
## -4.32273721 -0.56636844 -0.04703011  0.52025097  2.58014393 
## 
## Number of Observations: 195
## Number of Groups: 39

Model selection RI with interactions or RIAS

probably a reasonable assumptions for other linear model is an order 2 polynomial with trt poly order 2 fixe effect interaction interaction to account for change in simple curvature see longitudin l graph

The following model frame is one selected:

On other way a RIAS without/with intercations might be envision Despite the high value of random time effect (Slope effect) the path line graph of id dosen’t show up much variability agaisnt one individual at somme exeption.

Sevreals variance epress and some confusion might occur

Withi between longitidunal

It is ofeten reported that longitudinal variance is increasing heneec a CPS strucute of the cov matrix is innapropriate This fact can be explain by the fact that a dissference of two trt that one is working therefor an interaction with time is present the variance become increasing even seometimes xplosive( GArch model i.e). It is even proven in the ZDZt matrix of a RIAS model the variance in time slope become a squared or even 4 power of time spacing.

A time is a dynamic variable and is always a within subject factor.

Howvere a random intercept might be considered as a between id factor as by definition it is a varaince around the mean intercept and therefore characterize a between subject difference from mean population.

Maybe that accounting anfcfixed effect interaction plus a RAIAS model is too much?: It depends on the reserach thematic: For a prediction the advantage of individual time slope is an advantages If the study remain at the sample-population under study a simple RI with fixed effect interaction might more manageable!

Of course teh RIAS model account for the slope of the two treated groups as one random slope: But looking to the grapph variance is explained by group treatemnt: If you are willing to account exactly for group effect in random ness you therefore has to add an random interaction term slope with trt lme(Score_{time+trt,random=}trt*time+time|id,data=new,control = lmeControl(maxIter = 2000 However no convergence is found with so low data points-

Another way to do is to consider time a ordered factor:

## 
## Call:
## lm(formula = Score ~ ordered(time) * trt, data = new)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.5789  -1.4211  -0.1579   1.5789   4.5500 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          109.58947    0.22898 478.600   <2e-16 ***
## ordered(time).L      -12.13316    0.51201 -23.697   <2e-16 ***
## ordered(time).Q       -1.33631    0.51201  -2.610   0.0098 ** 
## ordered(time).C       -0.28294    0.51201  -0.553   0.5812    
## ordered(time)^4        0.38373    0.51201   0.749   0.4545    
## trtB                   4.49053    0.31975  14.044   <2e-16 ***
## ordered(time).L:trtB   8.22775    0.71499  11.508   <2e-16 ***
## ordered(time).Q:trtB   1.77729    0.71499   2.486   0.0138 *  
## ordered(time).C:trtB   0.50430    0.71499   0.705   0.4815    
## ordered(time)^4:trtB  -0.09688    0.71499  -0.135   0.8924    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.232 on 185 degrees of freedom
## Multiple R-squared:  0.8176, Adjusted R-squared:  0.8087 
## F-statistic: 92.11 on 9 and 185 DF,  p-value: < 2.2e-16

Result Still in accordance what was presented before : Linear if the main trend and a order 2 might be adequate for more precision on predictions.

REFERENCES

Chen, D. (., Peace, K. E. (2011). Clinical Trial Data Analysis Using R. États-Unis: CRC Press.

Large clinical trials have shown that combined treatment with reserpine plus a thiazide diuretic reduces mortality of people with hypertension..↩︎
Back on tracking the history of this RCT (part of the job) it is not clear from source which treatment was tested either Resepinic or betaB…The intention here is to demonstrate the technical statistical aspect with repeated data not a biophysical full proof of a drug per se.↩︎
Missing value will be replace by a Propensity score matching approach in this slight unbalanced design to make an repeated Anova and result caompared.↩︎
Naive because it doesn’t take into account repeated corrrelated measures hence partial information only Use LMM or GEE↩︎
[Naive model] doesnt take accout of RM↩︎

Longitudinal regression and Baseline inclusion

Mudryjm

2025-11-21

GOAL

Rct and DATA

Know your Data: Systematic review

Data summary

Baseline B0 value

Within and Between variability

Analysis :From Baseline to a 4 Months time point [Naive⁴]

Some comparison at Baseline difference:

Looking for the effect

Inference: T.Test

Correlations

Linear model with wave time points⁵

LINEAR MODEL TRT VS TRT BASELINE

Now lets explore an DOD Y|X = (BP4 BPbaseline) “like” regression with and without baseline in the X covariate

LINEAR MODEL with Y as Differnece from baseline

With-without baseline inclusion on covariate Xi also!

LONGITUDINAL DATA ANALYSIS

Now including a baseline as fixed covariate (long format R)

REFERENCES

id	trt	baseline	bp1	bp2	bp3	bp4	age	sex
2	A	116	113	112	103	101	51	M
3	A	119	115	113	104	98	48	F
4	A	115	113	112	109	101	42	F
5	A	116	112	107	104	105	49	M
6	A	117	112	113	104	102	47	M
7	A	118	111	100	109	99	50	F
8	A	120	115	113	102	102	61	M
9	A	114	112	113	109	103	43	M
10	A	115	113	108	106	97	51	M
11	A	117	112	110	109	101	47	F
12	A	116	115	113	109	102	45	M
13	A	119	117	110	106	104	54	F
14	A	118	115	113	102	99	52	M
15	A	115	112	108	105	102	42	M
16	A	114	111	111	107	100	44	F
17	A	117	114	110	108	102	48	M
18	A	120	115	113	107	103	63	F
19	A	114	113	109	104	100	41	M
20	A	117	115	113	109	101	51	M
21	B	114	115	113	111	113	39	M
22	B	116	114	114	109	110	40	F
23	B	114	115	113	111	109	39	F
24	B	114	115	113	114	115	38	M
25	B	116	113	113	109	109	39	F
26	B	114	115	114	111	110	41	M
27	B	119	118	118	117	115	56	F
28	B	118	117	117	116	112	56	M
29	B	114	113	113	109	108	38	M
30	B	120	115	113	113	113	57	M
31	B	117	115	113	114	115	47	F
32	B	118	114	112	109	110	48	M
33	B	121	119	117	114	115	61	F
34	B	116	115	116	114	111	49	M
35	B	118	118	113	113	112	52	M
36	B	119	115	115	114	111	55	F
37	B	116	114	113	109	109	45	F
38	B	116	115	114	114	112	42	M
39	B	117	115	113	114	115	49	F
40	B	118	114	114	114	115	50	F

id	trt	baseline	bp1	bp2	bp3	bp4	age	sex
2	A	116	113	112	103	101	51	M
3	A	119	115	113	104	98	48	F
4	A	115	113	112	109	101	42	F
5	A	116	112	107	104	105	49	M
6	A	117	112	113	104	102	47	M
7	A	118	111	100	109	99	50	F
8	A	120	115	113	102	102	61	M
9	A	114	112	113	109	103	43	M
10	A	115	113	108	106	97	51	M
11	A	117	112	110	109	101	47	F
12	A	116	115	113	109	102	45	M
13	A	119	117	110	106	104	54	F
14	A	118	115	113	102	99	52	M
15	A	115	112	108	105	102	42	M
16	A	114	111	111	107	100	44	F
17	A	117	114	110	108	102	48	M
18	A	120	115	113	107	103	63	F
19	A	114	113	109	104	100	41	M
20	A	117	115	113	109	101	51	M
21	B	114	115	113	111	113	39	M
22	B	116	114	114	109	110	40	F
23	B	114	115	113	111	109	39	F
24	B	114	115	113	114	115	38	M
25	B	116	113	113	109	109	39	F
26	B	114	115	114	111	110	41	M
27	B	119	118	118	117	115	56	F
28	B	118	117	117	116	112	56	M
29	B	114	113	113	109	108	38	M
30	B	120	115	113	113	113	57	M
31	B	117	115	113	114	115	47	F
32	B	118	114	112	109	110	48	M
33	B	121	119	117	114	115	61	F
34	B	116	115	116	114	111	49	M
35	B	118	118	113	113	112	52	M
36	B	119	115	115	114	111	55	F
37	B	116	114	113	109	109	45	F
38	B	116	115	114	114	112	42	M
39	B	117	115	113	114	115	49	F
40	B	118	114	114	114	115	50	F

Longitudinal regression and Baseline inclusion

Mudryjm

2025-11-21

GOAL

Rct and DATA

Know your Data: Systematic review

Data summary

Baseline B0 value

Within and Between variability

Analysis :From Baseline to a 4 Months time point [Naive4]

Some comparison at Baseline difference:

Looking for the effect

Inference: T.Test

Correlations

Linear model with wave time points5

LINEAR MODEL TRT VS TRT BASELINE

Now lets explore an DOD Y|X = (BP4 BPbaseline) “like” regression with and without baseline in the X covariate

LINEAR MODEL with Y as Differnece from baseline

With-without baseline inclusion on covariate Xi also!

LONGITUDINAL DATA ANALYSIS

Now including a baseline as fixed covariate (long format R)

REFERENCES

Analysis :From Baseline to a 4 Months time point [Naive⁴]

Linear model with wave time points⁵

id	trt	baseline	bp1	bp2	bp3	bp4	age	sex
2	A	116	113	112	103	101	51	M
3	A	119	115	113	104	98	48	F
4	A	115	113	112	109	101	42	F
5	A	116	112	107	104	105	49	M
6	A	117	112	113	104	102	47	M
7	A	118	111	100	109	99	50	F
8	A	120	115	113	102	102	61	M
9	A	114	112	113	109	103	43	M
10	A	115	113	108	106	97	51	M
11	A	117	112	110	109	101	47	F
12	A	116	115	113	109	102	45	M
13	A	119	117	110	106	104	54	F
14	A	118	115	113	102	99	52	M
15	A	115	112	108	105	102	42	M
16	A	114	111	111	107	100	44	F
17	A	117	114	110	108	102	48	M
18	A	120	115	113	107	103	63	F
19	A	114	113	109	104	100	41	M
20	A	117	115	113	109	101	51	M
21	B	114	115	113	111	113	39	M
22	B	116	114	114	109	110	40	F
23	B	114	115	113	111	109	39	F
24	B	114	115	113	114	115	38	M
25	B	116	113	113	109	109	39	F
26	B	114	115	114	111	110	41	M
27	B	119	118	118	117	115	56	F
28	B	118	117	117	116	112	56	M
29	B	114	113	113	109	108	38	M
30	B	120	115	113	113	113	57	M
31	B	117	115	113	114	115	47	F
32	B	118	114	112	109	110	48	M
33	B	121	119	117	114	115	61	F
34	B	116	115	116	114	111	49	M
35	B	118	118	113	113	112	52	M
36	B	119	115	115	114	111	55	F
37	B	116	114	113	109	109	45	F
38	B	116	115	114	114	112	42	M
39	B	117	115	113	114	115	49	F
40	B	118	114	114	114	115	50	F