# Marginal Effects in Two-Part Fractional Models

As shown in “Two-Part Fractional Model” posted on 09/25/2012, sometimes it might be beneficial to model fractional outcomes in the range of [0, 1] with composite models, e.g. a two-part model, especially when there are a non-trivial number of boundary outcomes. However, the marginal effect of X in a two-part model is not as straightforward to calculate as in a one-part model shown in “Marginal Effects in Tobit Models” posted on 10/06/2012.

In the demonstration below, I will show how to calculate the marginal effect of X in a two-part model with a similar logic shown in McDonald and Moffitt decomposition.

```proc nlmixed data = one tech = congra maxiter = 1000;
parms b10 = -9.3586 b11 = -0.0595 b12 =  1.7644 b13 =  0.5994 b14 = -2.5496
b15 = -0.0007 b16 = -0.0011 b17 = -1.6359
b20 =  0.3401 b21 =  0.0274 b22 =  0.1437 b23 =  0.0229 b24 =  0.4656
b25 =  0.0011 b26 =  0.0021 b27 =  0.1977  s  =  0.2149;
logit_xb = b10 + b11 * x1 + b12 * x2 + b13 * x3 + b14 * x4 +
b15 * x5 + b16 * x6 + b17 * x7;
nls_xb = b20 + b21 * x1 + b22 * x2 + b23 * x3 + b24 * x4 +
b25 * x5 + b26 * x6 + b27 * x7;
p1 = 1 / (1 + exp(-logit_xb));
p2 = 1 / (1 + exp(-nls_xb));
if y = 0 then ll = log(1 - p1);
else ll = log(p1) + log(pdf('normal', y, p2, s));
model y ~ general(ll);
predict logit_xb out = out_1 (rename = (pred = part1_xb) keep = _id_ pred y);
predict p1       out = out_2 (rename = (pred = part1_p)  keep = _id_ pred);
predict nls_xb   out = out_3 (rename = (pred = part2_xb) keep = _id_ pred);
predict p2       out = out_4 (rename = (pred = part2_p)  keep = _id_ pred);
run;

data out;
merge out_1 out_2 out_3 out_4;
by _id_;

margin1_part1 = (exp(part1_xb) / ((1 + exp(part1_xb)) ** 2) * -0.05948) * part2_p;
margin1_part2 = (exp(part2_xb) / ((1 + exp(part2_xb)) ** 2) * -0.01115) * part1_p;
x1_margin = margin1_part1 + margin1_part2;

margin2_part1 = (exp(part1_xb) / ((1 + exp(part1_xb)) ** 2) * 1.7645) * part2_p;
margin2_part2 = (exp(part2_xb) / ((1 + exp(part2_xb)) ** 2) * -0.4363) * part1_p;
x2_margin = margin2_part1 + margin2_part2;

margin3_part1 = (exp(part1_xb) / ((1 + exp(part1_xb)) ** 2) * 0.5994) * part2_p;
margin3_part2 = (exp(part2_xb) / ((1 + exp(part2_xb)) ** 2) * -0.1139) * part1_p;
x3_margin = margin3_part1 + margin3_part2;

margin4_part1 = (exp(part1_xb) / ((1 + exp(part1_xb)) ** 2) * -2.5496) * part2_p;
margin4_part2 = (exp(part2_xb) / ((1 + exp(part2_xb)) ** 2) * -2.8755) * part1_p;
x4_margin = margin4_part1 + margin4_part2;

margin5_part1 = (exp(part1_xb) / ((1 + exp(part1_xb)) ** 2) * -0.00071) * part2_p;
margin5_part2 = (exp(part2_xb) / ((1 + exp(part2_xb)) ** 2) * 0.004091) * part1_p;
x5_margin = margin5_part1 + margin5_part2;

margin6_part1 = (exp(part1_xb) / ((1 + exp(part1_xb)) ** 2) * -0.00109) * part2_p;
margin6_part2 = (exp(part2_xb) / ((1 + exp(part2_xb)) ** 2) * -0.00839) * part1_p;
x6_margin = margin6_part1 + margin6_part2;

margin7_part1 = (exp(part1_xb) / ((1 + exp(part1_xb)) ** 2) * -1.6359) * part2_p;
margin7_part2 = (exp(part2_xb) / ((1 + exp(part2_xb)) ** 2) * -0.1666) * part1_p;
x7_margin = margin7_part1 + margin7_part2;
run;

proc means data = _last_ mean;
var x:;
run;
/*
Variable                 Mean

x1_margin          -0.0039520
x2_margin           0.0739847
x3_margin           0.0270673
x4_margin          -0.3045967
x5_margin         0.000191015
x6_margin        -0.000533998
x7_margin          -0.1007960
*/
```