Need help to decide which statistical test to use

#1
Which statistical analysis should I use to measure difference in length of stay for a specific surgery (knee replacement and Hip replacement) among groups of patients (race groups, insurance types, different age groups etc.)?
 
#3
If you have age as 1, 2, 3,...,75... I would use a multiple regression model so you can leave this as a quantitative variable rather than inappropriately making age (quantitative) into a discrete variable. You can also directly get effect size estimates in the estimated betas (even if you use age as a group).

If you're concerned about normality or other assumptions for ANOVA, you may consider a nonparametric rank regression (just understand you're looking at mean rank for length of stay vs mean length of stay)...there are equivalencies to recognize when using a regression model with a quantitative response and only categorical variables vs using the "ANOVA." You can derive the same answers, but seeing the connection can provide a bit of flexibility and extra interpretable output.

Otherwise, I would agree with the two-way ANOVA.
 
#5
There seems to be four factors (knee replacement and Hip, race groups, insurance types, different age groups) where age could be a covariate = a regression variable.
 
#6
There seems to be four factors (knee replacement and Hip, race groups, insurance types, different age groups) where age could be a covariate = a regression variable.
Good point-- I interpreted the OP as wanting to do a 2 way ANOVA, essentially. If all those are reasonably assumed to influence the outcome, then including all in an a*b*c*d factorial may be a good idea (or just a regression to keep age quantitative, as we mentioned).
 
#8
I would guess that this is not a balanced design (They have hardly randomized people to different treatments) so I guess that the full factorial can not be estimated. But I guess that the main effects can be estimated.
Yes, at least a main effects.

Maybe OP can give us some descriptive statistics so we can see the number of observations in each group so we can get a better idea on the project.