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I'd like to get statistical support for hypotheses concerning the effects of independent variables $D,CF,CT,P,H,LA,LP$ on $C$. For regressor sets $\{D,CF,CT,P,LA,LP\}$ and $\{D,CF,CT,H,LA,LP\}$ all the independent variables are statistically significant:

Call:
clogit(C ~ D + CF + CT + P + LA + LP + strata(Case), clog_data)

                       coef exp(coef)  se(coef)      z       p
D                 -5.91e-03  9.94e-01  7.21e-05 -81.94 < 2e-16
CF                 3.78e-03  1.00e+00  8.10e-04   4.67 3.0e-06
CT                 2.60e-04  1.00e+00  5.18e-05   5.02 5.2e-07
P                  6.07e-01  1.84e+00  3.32e-02  18.32 < 2e-16
LA                 8.71e-02  1.09e+00  2.41e-02   3.62 0.00029
LP                -8.93e-02  9.15e-01  1.83e-02  -4.89 1.0e-06

Likelihood ratio test=11528  on 6 df, p=0
n= 71403, number of events= 12339 

Call:
clogit(C ~ D + CF + CT + H + LA + LP + strata(Case), clog_data)

                       coef exp(coef)  se(coef)      z       p
D                 -5.80e-03  9.94e-01  7.15e-05 -81.15 < 2e-16
CF                 7.87e-03  1.01e+00  7.73e-04  10.17 < 2e-16
CT                 3.13e-04  1.00e+00  5.10e-05   6.13 8.9e-10
H                 -9.47e-02  9.10e-01  2.01e-02  -4.71 2.5e-06
LA                 9.49e-02  1.10e+00  2.39e-02   3.97 7.2e-05
LP                -5.75e-02  9.44e-01  1.81e-02  -3.18  0.0015

Likelihood ratio test=11206  on 6 df, p=0
n= 71403, number of events= 12339 

For the full set of regressors $\{D,CF,CT,P,H,LA,LP\}$, one of the independent variables $(H)$ is not statistically significant

Call:
clogit(C ~ D + CF + CT + P + H + LA + LP + strata(Case), clog_data)

                       coef exp(coef)  se(coef)      z       p
D                 -5.90e-03  9.94e-01  7.24e-05 -81.53 < 2e-16
CF                 3.73e-03  1.00e+00  8.16e-04   4.57 4.8e-06
CT                 2.61e-04  1.00e+00  5.18e-05   5.03 4.9e-07
P                  6.03e-01  1.83e+00  3.40e-02  17.74 < 2e-16
H                 -1.10e-02  9.89e-01  2.08e-02  -0.53 0.59625
LA                 8.68e-02  1.09e+00  2.41e-02   3.61 0.00031
LP                -8.95e-02  9.14e-01  1.83e-02  -4.89 9.9e-07

Likelihood ratio test=11528  on 7 df, p=0
n= 71403, number of events= 12339 

Are my hypotheses statistically supported?

Update1: Correlation matrix

            Case     D     CF    CT    H     P     LP    LA     C
  Case      1.00    -0.01  0.00 -0.01 -0.01 -0.01  0.00 -0.01   0.00
  D        -0.01     1.00 -0.05  0.05  0.07  0.12  0.06  0.04  -0.35
  CF        0.00    -0.05  1.00 -0.18 -0.28  0.40  0.19  0.17   0.08
  CT       -0.01     0.05 -0.18  1.00  0.07 -0.03 -0.07 -0.04   0.00
  H        -0.01     0.07 -0.28  0.07  1.00 -0.31 -0.12 -0.12  -0.06
  P        -0.01     0.12  0.40 -0.03 -0.31  1.00  0.24  0.21   0.06
  LP        0.00     0.06  0.19 -0.07 -0.12  0.24  1.00  0.81   0.00
  LA       -0.01     0.04  0.17 -0.04 -0.12  0.21  0.81  1.00   0.01
  C         0.00    -0.35  0.08  0.00 -0.06  0.06  0.00  0.01   1.00

Update2: VIF

                      GVIF Df GVIF^(1/(2*Df))
D                 1.027972  1        1.013890
CF                1.210073  1        1.100033
CT                1.049540  1        1.024471
P                 1.245100  1        1.115841
H                 1.124559  1        1.060452
LA                2.684257  1        1.638370
LP                2.749423  1        1.658138
strata(Case)      3.470383  0             Inf
Warning message:
In vif.default(clogit(C ~ D + CF + CT + P + H +  :
No intercept: vifs may not be sensible.
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    $\begingroup$ What are your variables? In particular, what are H and P? It looks like they might be colinear. $\endgroup$ – Peter Flom Nov 2 '17 at 12:42
  • $\begingroup$ Thanks. All independent variables are continuous factors for choice $C$ which is binomial. Correlation between $H$ and $P$ is -0.31 $\endgroup$ – 8k14 Nov 2 '17 at 13:12
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    $\begingroup$ The problem is not limited to $H$ and $P$, as evidenced by the substantial changes in the estimates of $CF$ and $LP$ as $H$ and $P$ are added. $\endgroup$ – whuber Nov 2 '17 at 13:17
  • $\begingroup$ You didn't answer my question. Also, colinearity is not the same as correlation. $\endgroup$ – Peter Flom Nov 2 '17 at 21:02
  • $\begingroup$ @Peter Flom I'm sorry. What would you like to know about $H$ and $P$? $\endgroup$ – 8k14 Nov 3 '17 at 5:35
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With 12000+ events you are far from overfitting, so using the full set of regressors allows you to minimize potential problems from omitted-variable bias. In that full model, variable H does not pass the standard $p < 0.05$ test of statistical significance, while all the others do (even if you feel compelled to correct for multiple comparisons for the 7 coefficients). That's the model to focus on.

The results from the models that omit one of H and P aren't surprising, given the correlation matrix and the coefficients found for those two variables. They have a reasonably large negative correlation and regression coefficients of opposite signs. When you omit P from the regression, H then is able to pick up some of the influence of P even if its direct relation to outcome is minimal. That's consistent with omitted-variable bias in the regression omitting P. The simplest explanation is that H isn't closely related to outcome; it only appears to be if you ignore P, which is highly related both to outcome and to H.

VIF tells you how much the variance of the estimated coefficient might be inflated by correlations with other variables. A low VIF doesn't rule out omitted-variable bias.

This page is one of many on this site that discuss this general issue.

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  • $\begingroup$ Thanks a lot for the detailed answer. I think now the picture is clear for me. Back to my initial question: effects for all the independent variables except $H$ are statistically supported and effect for $H$ is not, right? $\endgroup$ – 8k14 Nov 2 '17 at 17:19
  • $\begingroup$ @8k14 Yes, that would be my interpretation of the analyses that you presented. $\endgroup$ – EdM Nov 2 '17 at 17:20
  • $\begingroup$ Thanks. On the other hand if I'm presented only with the second regression ($P$ omitted) I have solid reasons to believe that the hypothesis for $H$ is supported, right? $\endgroup$ – 8k14 Nov 2 '17 at 17:24
  • $\begingroup$ If you had no information about P and its relation both to outcome and to H then you might claim to have found statistical evidence for a role of H based on that limited model. That claim, however, would not be a good model of reality; with omitted-variable bias a "statistically significant" result can be grossly erroneous. $\endgroup$ – EdM Nov 2 '17 at 17:42
  • $\begingroup$ Thanks again. How can I know that this limited model is not a good model of reality? Are there any statistical methods for that or it's only in the nature of the process? $\endgroup$ – 8k14 Nov 2 '17 at 17:53
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You should check VIF because probably You have variables P and H highly correlated, which broke assumption of the regression in 3rd model.

If a information from one variable is incorporated to the model the second didn't has a chance to be significant.

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  • $\begingroup$ The high correlation between P and H doesn't break any assumptions about the logistic regression, but as you point out it does make it difficult to separate out any individual contributions of those 2 variables to the outcome. $\endgroup$ – EdM Nov 2 '17 at 13:45
  • $\begingroup$ Thanks. VIFs and correlation matrix are now included in my question. I may be wrong but I don't see collinearity here. $\endgroup$ – 8k14 Nov 2 '17 at 14:40

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