May be not entirely related to this question, here is a general strategy for perform model building (to your question on chat box...)

1. Always keep the known confounders in the model regardless of their significance. For e.g. Age which is a surrogate of immunity affects the disease outcome [This may not be true for all diseases but will depend on the scientific knowledge. You should be prepared to get out your statistical purity and respect the science where there is a sound reason. Hence, always keep the known confounders in the model, significant or not significant.

2. Use all those univariate risk factors significant at 10% or 20% level for model building [Although valid, I prefer likelihood ratio test..]

3. Use a likelihood ratio test (LRT) to determine which variables to include in the model [Don't use wald p-value of the model coefficients!!!]. A LRT tests how much the model is improved by adding the variable.

4. If a LRT is not significant i.e. if the variable doesn't significantly improve the model fit, check the effect of dropping the variable on the remainder model coefficients.

5. For the final multivariate model, perform residual analysis, robustness of model fit. Check the effect of adding of all the variables you had left out to see if it alters the results. If possible, check for interaction for the variables in your final model.

Once you've come up with a final model, perform cross-validation, robustness testing, sensitivity analysis...

Main thing to check with your model: Does your model makes sense? Are they realistic?

I have seen that previous studies first have run bivariate chi-square tests and then entered only those variables with chi-square P values < 0.1 or smaller than 0.15 into the model.

It is an acceptable and widely used method to include only those univariate risk factors singificant at 10% or 20% level. However, Likelihood ratio test is preferred, as said before.

A Likelihood Ratio is answer to almost any question

However, you have to include scientific knowledge to the model building process.