How to proceed with my multiple logistic regression?

Question

I am conducting a univariate logistic regression to determine factors that predict the success of a surgery. The significant variables were then added to a multiple-regression model. However, the problem is that when I add more than one variable (with any combination), the results become non-significant. I performed backward regression, and the two remaining variables are also non-significant.

Here are the results of the multiple regression I used in R:

Call:
glm(formula = Procedure ~ Zscore.of.the.LVEDD + Left.ventricular.end.diastolic.dimension.in.long.axis.view + 
    Left.ventricula.end.diatolic.dimension.in.4.chamber.view + 
    Left.ventricular.end.systolic.dimension.in.4.chamber.view + 
    Zscore.of.mitral.valve.size, family = binomial(), data = data)
Deviance Residuals: 
    Min       1Q   Median       3Q      Max

-2.4556  -0.4070   0.4628   0.7305   1.2196
Coefficients:
                                                           Estimate Std. Error z value Pr(>|z|)
(Intercept)                                                 5.20311    7.15494   0.727    0.467
Zscore.of.the.LVEDD                                         0.35359    0.29638   1.193    0.233
Left.ventricular.end.diastolic.dimension.in.long.axis.view  0.29624    0.46897   0.632    0.528
Left.ventricula.end.diatolic.dimension.in.4.chamber.view   -0.22014    0.47514  -0.463    0.643
Left.ventricular.end.systolic.dimension.in.4.chamber.view   0.03214    0.54687   0.059    0.953
Zscore.of.mitral.valve.size                                 0.70673    0.46260   1.528    0.127
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 39.429  on 29  degrees of freedom

Residual deviance: 27.898  on 24  degrees of freedom
AIC: 39.898
Number of Fisher Scoring iterations: 5

And here are the results of the backward regression:

Start:  AIC=39.9
Procedure ~ Zscore.of.the.LVEDD + Left.ventricular.end.diastolic.dimension.in.long.axis.view + 
    Left.ventricula.end.diatolic.dimension.in.4.chamber.view + 
    Left.ventricular.end.systolic.dimension.in.4.chamber.view + 
    Zscore.of.mitral.valve.size
                                                         Df Deviance    AIC


Left.ventricular.end.systolic.dimension.in.4.chamber.view   1   27.902 37.902
Left.ventricula.end.diatolic.dimension.in.4.chamber.view    1   28.116 38.116
Left.ventricular.end.diastolic.dimension.in.long.axis.view  1   28.304 38.304
Zscore.of.the.LVEDD                                         1   29.488 39.488

<none>                                                            27.898 39.898

Zscore.of.mitral.valve.size                                 1   30.723 40.722

Step:  AIC=37.9
Procedure ~ Zscore.of.the.LVEDD + Left.ventricular.end.diastolic.dimension.in.long.axis.view + 
    Left.ventricula.end.diatolic.dimension.in.4.chamber.view + 
    Zscore.of.mitral.valve.size
                                                         Df Deviance    AIC


Left.ventricula.end.diatolic.dimension.in.4.chamber.view    1   28.148 36.148
Left.ventricular.end.diastolic.dimension.in.long.axis.view  1   28.305 36.305
Zscore.of.the.LVEDD                                         1   29.784 37.784

<none>                                                            27.902 37.902

Zscore.of.mitral.valve.size                                 1   30.951 38.951

Step:  AIC=36.15
Procedure ~ Zscore.of.the.LVEDD + Left.ventricular.end.diastolic.dimension.in.long.axis.view + 
    Zscore.of.mitral.valve.size
                                                         Df Deviance    AIC


Left.ventricular.end.diastolic.dimension.in.long.axis.view  1   28.307 34.307
Zscore.of.the.LVEDD                                         1   30.058 36.058

<none>                                                            28.148 36.148

Zscore.of.mitral.valve.size                                 1   31.307 37.307

Step:  AIC=34.31
Procedure ~ Zscore.of.the.LVEDD + Zscore.of.mitral.valve.size
                          Df Deviance    AIC

<none>                             28.307 34.307

Zscore.of.mitral.valve.size  1   32.291 36.291
Zscore.of.the.LVEDD          1   32.352 36.352

It is worth mentioning that these variables are highly correlated. How to properly interpret my results in this scenario? Also, please note that the sample size is small (events = 19 and no event = 11).

Answer 1

Regarding your first point:

However, the problem is that when I add more than one variable (with any combination), the results become non-significant. I performed backward regression, and the two remaining variables are also non-significant.

You need to step back for a second and forget about statistical significance. Rather than using $p$ values to determine your beliefs, you should instead be looking at your data and how it may/may not match expectation. These are some more important questions to ask yourself:

• What does the data tell me visually? Is there an actual effect? Are there hidden relationships? (hint: you need to plot your data and see what's going on).
• Does past research or my theory explain this in some way?
• Is some causal pathway or otherwise spurious relationship influencing the results?
• Are there problems with my data (e.g. poor reliability of measures, erroneous outliers) that effect the results?

In the comments of another answer here you noted:

Basically, I need a way to prove that they are significant, but for a reason (point #1) they are not significant in multivariate regression, OR they are actually non-significant, and I need to know how to prove this.

Let us say that you do not uncover any serious errors, visualization shows that the effects are what you expect, and the results support/refute your theory in a logical way. Surprising findings can be just as important and interesting if you have a good justification for why. What does this mean for your research? Does it challenge some old ways of thinking? I would approach it from this perspective, though doing so cautiously due to sample size constraints (if you are dedicated to $p$ values at least).

Some other important takeaways/solutions:

• Never use backwards regression or any form of stepwise regression. It capitalizes on chance findings. You can just search this site for the many warnings about this technique. A useful paper on this topic can be found here.
• I agree with Linus that it would be helpful to check the VIF rather than the correlations to see if that is part of the issue. There is a quick overview of VIF in this paper. The performance package in R allows you to check this easily by running check_collinearity(fit), as shown here.
• I disagree with Linus on the point of just gathering more observations after inspecting your data. This is considered by a good number of people a questionable research practice (QRP) and should be avoided. In short, it allows you to p-hack by collecting more people until your results look the way you want, which is unscientific. The opposite is just as true (selective stopping) when you stop collecting data as soon as your results are statistically significant.
• I would be very wary of the results here if I was a reviewer for a scientific journal given the tiny sample size. One approach you could consider is using a Bayesian regression instead, paying very careful attention to using well-calibrated priors. The book Statistical Rethinking by McElreath has a very accessible R-based approach to this topic. In short, any sample size is valid for Bayes, though with smaller sample sizes one must be especially cautious of the priors (otherwise Bayes can be just as destructive as NHST).

Answer 2

With a total of 30 cases (19 events and 11 non-events), your study is quite small. This can lead to a lack of statistical power, making it difficult to detect significant effects even if they exist. The small sample size also increases the risk of overfitting, especially when you include multiple predictors.

When your variables are highly correlated (multicollinearity), it can cause problems in multiple regression analysis. Multicollinearity can inflate the variance of the coefficient estimates, which can lead to a loss of statistical significance. You might want to check the Variance Inflation Factor (VIF) to assess the level of multicollinearity. Another thing you could try is to look into Lasso or Ridge regression which handles multicollinearity better and also work better in cases of a small sample size as you can penalize overfitting.

Also note that the p-Values of all your variables are fairly high. This means that there's a high chance none of your predictors (variables) actually matter (their estimates are likely to be zero or close to zero).

I'm sure as you're already aware that your sample size is extremely small I would first try to get more observations or try to augment them in some way to push significance up and the p-Values down to below 0.05.

Hope this helps!