Friday, February 03, 2006

st: How to best describe interaction between a dummy variable and a continuous one in logistic regression?

This is a statistics question rather than a Stata question. I am struggling with how to best describe (medical journal manuscript) an interaction effect.

The overall goal of the study is to describe the continuous variable 'zlog' as a predictor of 'outcome' and to determine the degree to which the association is independent of other variables. It is indeed independent of most of them, but there are two dummy variables for which interaction terms are significant and the odds ratio for zlog changes.

The question is: How do I describe/quantify the interaction in a succinct way? Do the odds ratios for the interaction terms have any intuitive meaning? I can see what is happening (sort of) by dropping observations based upon the dummy variable, but it is hard to describe quantitatively.

The two dummy variables that interact are different. In the first example, the odds ratio for the continuous variable increases when the either the observations with dummy==0 or dummy==1 are dropped. In the second case, dropping dummy==0 decreases the the OR for the continuous variable and dropping dummy==1 increases it.

*********************example #1: Dummy variable (romi) has increases OR for

the continuous variable (zlog) for both situations romi==0 and romi==1 . logistic outcome zlog

Logistic regression Number of obs = 20277 LR chi2(1) = 243.41 Prob > chi2 = 0.0000 Log likelihood = -3868.6247 Pseudo R2 = 0.0305

---------------------------------------------------------------------------- -- outcome | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+-------------------------------------------------------------- -- zlog | 2.080782 .0918129 16.61 0.000 1.908394 2.268742 ---------------------------------------------------------------------------- --

. logistic is_dead romi

Logistic regression Number of obs = 21236 LR chi2(1) = 278.92 Prob > chi2 = 0.0000 Log likelihood = -3928.3283 Pseudo R2 = 0.0343

---------------------------------------------------------------------------- -- outcome | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+-------------------------------------------------------------- -- romi | .2172724 .0241013 -13.76 0.000 .1748169 .2700384 ---------------------------------------------------------------------------- --

. logistic outcome zlog romi

Logistic regression Number of obs = 20277 LR chi2(2) = 553.98 Prob > chi2 = 0.0000 Log likelihood = -3713.3369 Pseudo R2 = 0.0694

---------------------------------------------------------------------------- -- outcome | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+-------------------------------------------------------------- -- zlog | 2.474006 .1195488 18.75 0.000 2.250448 2.719772 romi | .1786168 .0215241 -14.29 0.000 .1410422 .2262016 ---------------------------------------------------------------------------- --

. xi: logistic outcome i.romi*zlog i.romi _Iromi_0-1 (naturally coded; _Iromi_0 omitted) i.romi*zlog _IromXzlog_# (coded as above)

Logistic regression Number of obs = 20277 LR chi2(3) = 558.58 Prob > chi2 = 0.0000 Log likelihood = -3711.0395 Pseudo R2 = 0.0700

---------------------------------------------------------------------------- -- outcome | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+-------------------------------------------------------------- -- _Iromi_1 | .2093427 .0292553 -11.19 0.000 .1591856 .2753037 zlog | 2.350594 .1269027 15.83 0.000 2.114576 2.612955 _IromXzlog_1 | 1.295315 .1556027 2.15 0.031 1.023583 1.639185 ---------------------------------------------------------------------------- --

. preserve . drop if romi==1 (6413 observations deleted)

. logistic outcome zlog

Logistic regression Number of obs = 14823 LR chi2(1) = 230.33 Prob > chi2 = 0.0000 Log likelihood = -3335.5847 Pseudo R2 = 0.0334

---------------------------------------------------------------------------- -- outcome | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+-------------------------------------------------------------- -- zlog | 2.350594 .1269027 15.83 0.000 2.114576 2.612955 ---------------------------------------------------------------------------- --

. restore . drop if romi==0 (14823 observations deleted)

. logistic outcome zlog

Logistic regression Number of obs = 5454 LR chi2(1) = 91.85 Prob > chi2 = 0.0000 Log likelihood = -375.4548 Pseudo R2 = 0.1090

---------------------------------------------------------------------------- -- outcome | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+-------------------------------------------------------------- -- zlog | 3.04476 .3267399 10.38 0.000 2.467225 3.757486 ---------------------------------------------------------------------------- --

************************* ************************* example #2: dummy variable increases OR for zlog

if is_sec == 1 but increases it is is_sec==0

. logistic outcome zlog

Logistic regression Number of obs = 20277 LR chi2(1) = 243.41 Prob > chi2 = 0.0000 Log likelihood = -3868.6247 Pseudo R2 = 0.0305

---------------------------------------------------------------------------- -- outcome | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+-------------------------------------------------------------- -- zlog | 2.080782 .0918129 16.61 0.000 1.908394 2.268742 ---------------------------------------------------------------------------- --

.

. logistic outcome no_sec

Logistic regression Number of obs = 20277 LR chi2(1) = 42.08 Prob > chi2 = 0.0000 Log likelihood = -3969.2894 Pseudo R2 = 0.0053

---------------------------------------------------------------------------- -- outcome | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+-------------------------------------------------------------- -- no_sec | 1.543235 .101792 6.58 0.000 1.356085 1.756214 ---------------------------------------------------------------------------- --

.

. logistic outcome zlog no_sec

Logistic regression Number of obs = 20277 LR chi2(2) = 306.15 Prob > chi2 = 0.0000 Log likelihood = -3837.2553 Pseudo R2 = 0.0384

---------------------------------------------------------------------------- -- outcome | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+-------------------------------------------------------------- -- zlog | 2.146586 .0947182 17.31 0.000 1.968743 2.340495 no_sec | 1.71472 .1148813 8.05 0.000 1.503713 1.955335 ---------------------------------------------------------------------------- --

. xi: logistic outcome i.no_sec*zlog

i.no_sec _Ino_sec_0-1 (naturally coded; _Ino_sec_0 omitted)

i.no_sec*zlog _Ino_Xzlog_# (coded as above)

Logistic regression Number of obs = 20277 LR chi2(3) = 315.56 Prob > chi2 = 0.0000 Log likelihood = -3832.5466 Pseudo R2 = 0.0395

---------------------------------------------------------------------------- -- outcome | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+-------------------------------------------------------------- -- _Ino_sec_1 | 1.268187 .1531227 1.97 0.049 1.00094 1.606788 zlog | 2.396819 .1351902 15.50 0.000 2.145972 2.676988 _Ino_Xzlog_1 | .7573733 .0690372 -3.05 0.002 .6334612 .905524 ---------------------------------------------------------------------------- --

.

. preserve

. drop if no_sec==0

(13788 observations deleted)

. logistic outcome zlog

Logistic regression Number of obs = 6489 LR chi2(1) = 60.98 Prob > chi2 = 0.0000 Log likelihood = -1514.7069 Pseudo R2 = 0.0197

---------------------------------------------------------------------------- -- outcome | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+-------------------------------------------------------------- -- zlog | 1.815287 .129987 8.33 0.000 1.577587 2.088802 ---------------------------------------------------------------------------- --

. restore

. drop if no_sec==1

(6489 observations deleted)

. logistic outcome zlog

Logistic regression Number of obs = 13788 LR chi2(1) = 212.51 Prob > chi2 = 0.0000 Log likelihood = -2317.8397 Pseudo R2 = 0.0438

---------------------------------------------------------------------------- -- outcome | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+-------------------------------------------------------------- -- zlog | 2.396819 .1351902 15.50 0.000 2.145972 2.676988 ---------------------------------------------------------------------------- --

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