Joint cumulative distribution of X1 and X2

 

Proving law of the unconscious statistician (LOTUS) for discrete variable

 

Multiple conditions

From 

https://jonathanweisberg.org/vip/multiple-conditions.html#multiple-conditions

 9 Multiple Conditions | Odds & Ends (jonathanweisberg.org)

When to Adjust Alpha During Multiple Testing

https://www.qeios.com/read/QIH0Q1

When to Adjust Alpha During Multiple Testing - Article (v1) by Mark Rubin | Qeios

Counting number of duplicated cases by several variables

proc sort data=have(keep=key1 key2);
  by key1 key2;
run;
data want;
  do count=1 by 1 until(last.key2);
    set have;
    by key1 key2;
    end;
run;

A very clear introduction to B-SPLINES

 

MLE and REML in mixed models

 In mixed models, if use MLE, we have to assume that fixed effects are known, rather than being estimated from the data.  If we use REML (residual maximum likelihood) it will automatically adjust for the degree of freedom corresponding to estimated fixed effects. This is the key difference between MLE and REML in mixed models,   however, the difference will be very small when the sample size is big.