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when some of your coefficients in multivariate logistic regression model is negative while i know these variable have positive sign in univariate model, What should I do؟

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  • $\begingroup$ Welcome to the site! Could you describe what you mean by "univariate model"? And maybe describe the results you're getting in a little more detail. $\endgroup$ – smillig May 29 '15 at 8:49
  • $\begingroup$ This question is also answered at stats.stackexchange.com/questions/41633 (for general regression). $\endgroup$ – whuber Jun 4 '15 at 16:49
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This is a general phenomena caused by correlation between your independent variables. Here's a small example I constructed for you to experiment with. I demonstrated with a linear regression as the phenomena is easier to see pictorially in this case, but it also happens with any generalized linear model, including logistic regressions.

First, let's create a vector of random uniform values

x_1 <- runif(200, 0, 1)

and then construct another vector that is explicitly correlated with x_1

x_2 <- .5*x_1 + rnorm(200, 0, .25)

Since I forced a statstical dependency between x_1 and x_2, these random variables are correlated

cor(matrix(c(x_1, x_2), ncol=2))
          [,1]      [,2]
[1,] 1.0000000 0.5059403
[2,] 0.5059403 1.0000000

You can see this geometrically with a scatterplot

enter image description here

Now let's make a dependent variable that depends on both

y <- 3*x_1 - x_2 + rnorm(200, 0, .1)

This collection of three variables show the behavior that you are witnessing. Putting x_2 in a univariate model shows a positive coefficient

df <- data.frame(x_1=x_1, x_2=x_2, y=y)

# Univariate model
lm(y~x_2, data=df)

Call:
lm(formula = y ~ x_2, data = df)

Coefficients:
(Intercept)          x_2  
     1.1152       0.5061  

In fact, you can see that both x_1 and x_2 are positively correlated with y in a picture

enter image description hereenter image description here

But if I put them all together, I get recover the true negative coefficient for x_2!

# Multivariate model
lm(y ~ x_1 + x_2, data=df)

Call:
lm(formula = y ~ x_1 + x_2, data = df)

Coefficients:
(Intercept)          x_1          x_2  
   -0.01096      3.00800     -1.00790 
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  • $\begingroup$ Thanks for taking the time to provide this detailed example! $\endgroup$ – Negarev May 21 at 23:46
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The most common reason is that the remaining variables of the fitted model influence the changing of the sign!

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