paired data problem

[Cheng YuanLung]

I have data for the lipid profile before treatment, 4 weeks, 12 weeks after treatment.

Paired data analyses showed that LDL significantly increased 4 weeks after treatment, and no difference was noted between 4wk and 12 wk.

Could I conclude that the increase will mostly occur in 4 weeks after the treatment starts, and it isn't significant thereafter?

What is a better analysis for the conclusion I want?

Thank you.

[Carlo Lazzaro]

Cheng:
yes, you can, as you can see in the following toy-example:

. use "http://www.stata-press.com/data/r15/nlswork.dta"
. xtreg ln_wage i.year

Random-effects GLS regression                   Number of obs     =     28,534
Group variable: idcode                          Number of groups  =      4,711

R-sq:                                           Obs per group:
     within  = 0.1058                                         min =          1
     between = 0.0796                                         avg =        6.1
     overall = 0.0724                                         max =         15

                                                Wald chi2(14)     =    3198.67
corr(u_i, X)   = 0 (assumed)                    Prob > chi2       =     0.0000

------------------------------------------------------------------------------
     ln_wage |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        year |
         69  |   .0854251   .0123808     6.90   0.000     .0611592    .1096911
         70  |    .069889   .0115653     6.04   0.000     .0472215    .0925566
         71  |   .1196013   .0114272    10.47   0.000     .0972043    .1419983
         72  |   .1327835   .0117484    11.30   0.000      .109757    .1558099
         73  |   .1476736   .0113915    12.96   0.000     .1253467    .1700006
         75  |   .1609048   .0112586    14.29   0.000     .1388385    .1829712
         77  |   .2207951   .0112687    19.59   0.000     .1987089    .2428814
         78  |   .2596833   .0115128    22.56   0.000     .2371186     .282248
         80  |    .267724   .0116586    22.96   0.000     .2448735    .2905745
         82  |   .2852433   .0113995    25.02   0.000     .2629007    .3075859
         83  |   .3126372   .0115421    27.09   0.000     .2900152    .3352592
         85  |   .3653587   .0114383    31.94   0.000       .34294    .3877773
         87  |   .3813851   .0113703    33.54   0.000     .3590997    .4036706
         88  |   .4370689   .0113081    38.65   0.000     .4149054    .4592325
             |
       _cons |   1.426677   .0103611   137.70   0.000     1.406369    1.446984
-------------+----------------------------------------------------------------
     sigma_u |  .36928787
     sigma_e |  .30294584
         rho |  .59773703   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. test (69.year=70.year) (69.year=71.year)(70.year=71.year), mtest(bonferroni)

 ( 1)  69.year - 70.year = 0
 ( 2)  69.year - 71.year = 0
 ( 3)  70.year - 71.year = 0
       Constraint 3 dropped

---------------------------------------
       |        chi2     df       p
-------+-------------------------------
  (1)  |        1.76      1     0.5547 #
  (2)  |        8.66      1     0.0098 #
  (3)  |       22.55      1     0.0000 #
-------+-------------------------------
  all  |       23.45      2     0.0000
---------------------------------------
         # Bonferroni-adjusted p-values

.

[Cheng YuanLung]

Cheng:
I fail to get your concern.

Sorry that I didn't present my question well.
My question is:
May I have pairwise comparison results after performing panel data analyses? (wk4 vs wk12, wk4 vs wk24, w12 vs wk24, etc..)

Like the results from post-hoc analyses after performing ANOVA.

[Carlo Lazzaro]

Cheng:
I fail to get your concern.
if -hausman- outcome pointed you to -re-, this exactly what you have already did (see the results in #4).
I fail also to get what kind of problems you have with the Pregibon test.

[Cheng YuanLung]

-hausman- outcome suggests I go -re-.

The results look like those from regression analyses.

I can't tell the LDL increase is only significant between 0-4 week, but not 4-12 weeks, not 4-24 weeks, and not 12-24 weeks, from the results.

What do I miss?

I have problems with -Pregibon test- and will study about it later.

[Carlo Lazzaro]

Cheng:
assuming that -re- specification is the way to go and that you did not detect heteroskedasticity and/or autocorrelation in your data (if that were the case, impose the -robust- option on your standard errors), your conclusion would make sense if your regression model were well specified with only -i.time- as a predictor.
I think you have probably collected other predictors, such as age, geneder and comorbidities of patients enrolled in your study.
If that were the case, I would plug them in the right-hand side of your regression equation and see what happens with your outcome.
Besides, post estimation test play a relevant role in detecting model misspecification.
For instance, you can run the Pregibon test (see references and assumptions under -linktest- esntry in Stata .pdf manual) like in the following toy-example:

. use "http://www.stata-press.com/data/r15/nlswork.dta"
(National Longitudinal Survey.  Young Women 14-26 years of age in 1968)

. quietly xtreg ln_wage age i.year, fe

. estimates store fe

. quietly xtreg ln_wage age i.year, re

. estimates store re

*Hausman test*

. hausman fe re

                 ---- Coefficients ----
             |      (b)          (B)            (b-B)     sqrt(diag(V_b-V_B))
             |       fe           re         Difference          S.E.
-------------+----------------------------------------------------------------
         age |    .0125992     .0137208       -.0011216          .01004
        year |
         69  |    .0748621     .0744312        .0004309        .0098206
         70  |    .0478697     .0453659        .0025038        .0202541
         71  |    .0865577     .0819949        .0045629        .0303028
         72  |    .0856757     .0827461        .0029296        .0402525
         73  |    .0880069     .0840751        .0039318        .0503357
         75  |    .0778607     .0707387         .007122         .070056
         77  |     .108365     .1032639        .0051011        .0900953
         78  |    .1309518     .1279039        .0030479        .1005435
         80  |    .1142649      .108871        .0053939        .1202629
         82  |    .1090451      .098831        .0102141        .1403289
         83  |    .1211272     .1127655        .0083617        .1502649
         85  |    .1465637     .1380611        .0085026        .1703831
         87  |    .1382642     .1264818        .0117824        .1905725
         88  |    .1799741     .1640382        .0159359        .2042295
------------------------------------------------------------------------------
                           b = consistent under Ho and Ha; obtained from xtreg
            B = inconsistent under Ha, efficient under Ho; obtained from xtreg

    Test:  Ho:  difference in coefficients not systematic

                 chi2(15) = (b-B)'[(V_b-V_B)^(-1)](b-B)
                          =       81.61
                Prob>chi2 =      0.0000

. quietly xtreg ln_wage age i.year, fe

. predict fitted, xb
(24 missing values generated)

. g sq_fitted=fitted^2
(24 missing values generated)

*Pregibon test*

. xtreg ln_wage fitted sq_fitted, fe

Fixed-effects (within) regression               Number of obs     =     28,510
Group variable: idcode                          Number of groups  =      4,710

R-sq:                                           Obs per group:
     within  = 0.1086                                         min =          1
     between = 0.0962                                         avg =        6.1
     overall = 0.0845                                         max =         15

                                                F(2,23798)        =    1449.33
corr(u_i, Xb)  = 0.0519                         Prob > F          =     0.0000

------------------------------------------------------------------------------
     ln_wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      fitted |   4.791882   .4570443    10.48   0.000     3.896046    5.687718
   sq_fitted |  -1.125657   .1355632    -8.30   0.000    -1.391369   -.8599442
       _cons |    -3.1774   .3839526    -8.28   0.000    -3.929971   -2.424828
-------------+----------------------------------------------------------------
     sigma_u |  .40489607
     sigma_e |  .30248347
         rho |  .64180514   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(4709, 23798) = 8.78                 Prob > F = 0.0000

. test sq_fitted

 ( 1)  sq_fitted = 0

       F(  1, 23798) =   68.95
            Prob > F =    0.0000
.

As you can see, -hausman- outcome suggests to go -fe-, but the Pregibon test proves the regression model to be misspecified (that may happen for various reasons).

[Cheng YuanLung]

[QUOTE=Carlo Lazzaro;n1483220]Cheng:
looking at your outcome tables, what strikes me is the different number of observations between tests perfomed at 4 and 12 week after the treatment start.
I find difficult to get what's your original sample size and why it increased from the the first to the second measurement.
That said, you may want to consider a panel data regression (provided that the mesurements were taken on the same sample od patients):

xtreg LDL i.week

keep NO LDL LDL_4wk LDL_12wk LDL_SVR

rename LDL LDL0

rename LDL_4 LDL4

rename LDL_12 LDL12

rename LDL_SVR LDL24

reshape long LDL, i(NO) j(wk)

xtset NO

I think the results mean that LDL values at week4/12/24 are all significantly higher than the baseline.

Can the results tell me if the increase occurs more dramatically at the 4th week, and not after 4 weeks?

As we can see, the coefficient jumps to 14 in 4 weeks, and only slightly climbs to 17 at 24 weeks.

Thank you for your help, Carlo.

[Carlo Lazzaro]

Cheng:
looking at your outcome tables, what strikes me is the different number of observations between tests perfomed at 4 and 12 week after the treatment start.
I find difficult to get what's your original sample size and why it increased from the the first to the second measurement.
That said, you may want to consider a panel data regression (provided that the mesurements were taken on the same sample od patients):

xtreg LDL i.week

As per -hausman- test you can investigate whether -fe- or -re- specification fits you data better.

[Cheng YuanLung]

Below are the results by paired t test. Thank you for your help.
SVR is 12 week after treatment
LDL is before treatment
4wk is 4 wk after treatment