Marginal effect

Odd ratio 和relative risk缺点是不太直观，而且容易误解。

1. 例子

Trt组：The probability of death in the treatment group is 0.20

Placebo组：probability of death in a control group is 0.40. 

 是说治疗组死亡比值减少37.5%，而不是治疗组死亡概率减少37.5%。或者说Placebo组的死亡比值 高于治疗组2.67倍。注意：odd ratio这的%是值，而不是概率。和概率啥关系都没有。

相对风险是 0.2 / 0.4 = 50%. 治疗组死亡概率降低50%；注意：relative这的%是概率。

死亡风险降低一半，听起来效果极强，但如果把数据换成 0.0002 和 0.0004会得出相同的结论，这缺点就是反应不出强度。

2. Marginal effect

marginal effects is a way of presenting results as differences in probabilities

用概率差异表示结果。

  就是说用概率建模，而不是odd的对数。逻辑回归里面的系数就是odd的对数。

第一张图是说每多吸一单位的烟，低体重新生儿的odd增加4.59%；

第二张图是说每多吸一单位的烟，低体重新生儿的概率增加0.77%。吸10包烟就是增加百分之7.7%。

3.

data test;
    do a = 1 to 3 by 1;
        do y = 0,1;
        input freq @@;
        output;
        end;
    end;
    datalines;
    6 12
    35 23
    11 13
;

ods html;

proc logistic data=test;
class a / param=glm;
model y(event="1") = a / ;
lsmeans a / e ilink diff cl;
/*lsmestimate a [1,1] [-1,2] / cl;*/
store LogFit;
weight freq;
ods output coef=Coeffs;
run;

proc genmod data=test descending;
 class a;
 model y = a / dist=binomial link=identity;
 weight freq;
 lsmeans a / diff cl;
run;

proc catmod data=test;
    response 0 1;
    model y = a / param=ref clparm;
    weight freq;
    
    contrast "Prob Diff A1-A2" a 1 -1 / estimate=parm;
    contrast "Prob Diff A1-A3" a 1  0 / estimate=parm;
    contrast "Prob Diff A2-A3" a 0  1 / estimate=parm;
run; 
quit;

来自 Interpreting Model Estimates: Marginal Effects (ucdenver.edu)

37228 - Estimating differences in probabilities (marginal effects) with confidence interval (sas.com)