I have variable two dependent “total bone density “which measured in cm3 and “bone density percentage” which calculated by dividing dense bone area by total bone area.

I am trying to investigate if smoking is associated with "total bone density" and "bone density percent" after adjustment to alcohol and BMI. The dependents variables are positively skewed and also the residuals are not normally distributed. Therefore, I am using GLM.

Did I use the right link and family for both of dependent variables?

How should I interpret the coefficients for smoking variable?

How do calculate the adjusted mean and the 95% CI for bone density percentage and total bone density across the categories of the smoking.

My sample is around 1000 but the results for the GLM present for 418 only.

For total bone density
glm mean_tda age i.alcweek3 i.smoker3 i.cbmigroups3 ,family(poisson) link(log) vce(robust)
note: mean_tda has noninteger values

Iteration 0: log pseudolikelihood = -3997.8733
Iteration 1: log pseudolikelihood = -3932.7379
Iteration 2: log pseudolikelihood = -3932.6239
Iteration 3: log pseudolikelihood = -3932.6239

Generalized linear models No. of obs = 418
Optimization : ML Residual df = 408
Scale parameter = 1
Deviance = 6314.324466 (1/df) Deviance = 15.47629
Pearson = 8143.055288 (1/df) Pearson = 19.95847

Variance function: V(u) = u [Poisson]
Link function : g(u) = ln(u) [Log]

AIC = 18.86423
Log pseudolikelihood = -3932.623915 BIC = 3851.848

--------------------------------------------------------------------------------
| Robust
mean_tda | Coef. Std. Err. z P>|z| [95% Conf. Interval]
---------------+----------------------------------------------------------------
age | -.022811 .0070187 -3.25 0.001 -.0365674 -.0090546
|
alcweek3 |
three or less | -.0997237 .18797 -0.53 0.596 -.468138 .2686907
four to nine | .1863415 .2112141 0.88 0.378 -.2276306 .6003136
ten or more | .0019707 .1883379 0.01 0.992 -.3671648 .3711061
|
smoker3 |
former | -.1395443 .1253079 -1.11 0.265 -.3851433 .1060546
current | -.205274 .2045921 -1.00 0.316 -.6062672 .1957192
|
cbmigroups3 |
obese | -.4339761 .1483256 -2.93 0.003 -.7246888 -.1432633
overwight | -.2252917 .1470218 -1.53 0.125 -.5134491 .0628656
underweight | .0685373 .218682 0.31 0.754 -.3600715 .4971461
|
_cons | 4.245983 .468874 9.06 0.000 3.327007 5.164959
--------------------------------------------------------------------------------

For bone density percent
generate bd_pr=bd/100

glm bd_pr i.alcweek3 i.smoker3 i.cbmigroups3, link (logit) vce(robust)

Iteration 0: log pseudolikelihood = 215.01385
Iteration 1: log pseudolikelihood = 264.22253
Iteration 2: log pseudolikelihood = 271.92475
Iteration 3: log pseudolikelihood = 271.92981
Iteration 4: log pseudolikelihood = 271.92981

Generalized linear models No. of obs = 418
Optimization : ML Residual df = 409
Scale parameter = .0162899
Deviance = 6.662566651 (1/df) Deviance = .0162899
Pearson = 6.662566651 (1/df) Pearson = .0162899

Variance function: V(u) = 1 [Gaussian]
Link function : g(u) = ln(u/(1-u)) [Logit]

AIC = -1.258037
Log pseudolikelihood = 271.9298107 BIC = -2461.849

--------------------------------------------------------------------------------
| Robust
bd_pr | Coef. Std. Err. z P>|z| [95% Conf. Interval]
---------------+----------------------------------------------------------------
alcweek3 |
three or less | -.0583549 .1833873 -0.32 0.750 -.4177874 .3010775
four to nine | .1018567 .2133562 0.48 0.633 -.3163138 .5200273
ten or more | .106684 .1986481 0.54 0.591 -.2826591 .4960272
|
smoker3 |
former | -.2902414 .1494269 -1.94 0.052 -.5831127 .00263
current | -.4650027 .2777897 -1.67 0.094 -1.00946 .0794551
|
cbmigroups3 |
obese | -1.050255 .1678739 -6.26 0.000 -1.379282 -.7212283
overwight | -.572954 .1492834 -3.84 0.000 -.8655441 -.2803639
underweight | .5373612 .2693027 2.00 0.046 .0095377 1.065185
|
_cons | -1.494728 .1718149 -8.70 0.000 -1.831479 -1.157977
--------------------------------------------------------------------------------


. dataex bd mean_tda smoker3 cbmigroups3

----------------------- copy starting from the next line -----------------------
Code:
* Example generated by -dataex-. To install: ssc install dataex
clear
input float(bd mean_tda) byte(smoker3 cbmigroups3)
 16.365463  18.81128 1 1
  .2624936  .6569816 2 2
 10.473497  18.42222 2 3
  7.537117 16.728163 1 3
  3.052203  7.655602 1 2
 .55155325   .899969 1 2
  22.84368  25.15202 3 .
   52.8951   29.5906 2 3
 12.876885  7.408317 1 1
  21.18159 16.218298 1 1
  6.871165  7.035076 1 1
 16.478045 32.336216 2 2
  22.57178 26.838284 1 3
  .5757041  .7053235 2 3
  .4333925 1.0435233 1 2
  5.899091 12.323744 1 2
  6.383229 8.1892605 2 1
  .9931619  .9373725 1 3
  2.183749  1.460383 2 1
  18.61617 23.217287 2 1
 4.1581864  7.126842 1 3
 15.591076 11.341998 1 1
 19.108936  24.22366 1 1
  12.08951   41.5315 1 2
 34.814568 13.875312 1 4
 16.630676  24.59109 1 1
  18.88097 16.911541 2 3
  3.552727 3.7204306 1 2
   51.6785 36.629566 2 1
  2.426274 4.1825924 2 2
 2.0202687 2.3565574 3 1
 12.494894  11.57091 1 1
 2.4119585  3.584194 2 2
  16.57736  31.79659 2 2
   12.4815 24.271303 1 3
 14.112982 12.440155 1 1
  7.582244  6.519325 2 2
 16.261467  27.14514 1 3
 2.3648822  1.431556 1 3
 13.298046 13.003028 1 1
  1.938546 1.9457574 1 2
 19.626026  32.06744 2 1
  51.60948  26.92982 3 1
.030849867 .07045104 1 .
  32.83851 30.121876 . .
 3.6494505  4.137619 1 3
  9.280085  4.934806 2 1
  26.71291  23.61595 2 3
 .22426225 .31332225 2 3
 12.666924  20.42655 3 3
  2.127961 2.1698883 2 2
.016742319 .04563152 1 2
 30.939955  26.72437 2 1
  55.50521  24.20488 1 4
  .8922967 1.5079697 3 1
  9.707381  9.776858 2 1
  9.338659   13.2864 1 1
  9.825716 17.349289 2 2
  4.183007 3.0979486 1 1
   1.53075 1.8140126 1 2
  .3445268  .9344685 2 2
  25.31284  24.73643 2 3
 10.172702 22.785326 1 1
  7.473836 10.912245 1 3
 20.290724 15.662975 1 3
  15.29849 10.704066 1 1
 1.1178778 2.0278342 2 2
   .815134  .9314096 2 3
.011174798  .0213928 1 2
 1.4868008 2.2582278 1 2
  19.23166 28.618475 2 3
  9.470416  11.99108 1 3
  .4934571  .9587653 2 2
  6.995302 13.102035 1 2
  6.177939 16.233845 1 2
  55.24698 140.71783 1 3
  9.001475 10.394248 2 3
   14.2323  10.14222 1 1
  .8016296  .7042006 1 2
  20.81357 35.962902 1 3
  .4873792  .7268325 1 2
   2.02487 1.2113456 2 3
  20.75257 13.728292 2 1
 16.036518 33.274246 3 2
  1.456532 1.5249872 1 1
  7.016962  6.137817 1 3
  5.493763  5.105213 1 1
 4.5745535  3.688312 2 1
  5.293759  8.490792 3 2
  8.613267  21.78817 1 1
  3.941279 4.2355614 1 3
 3.5713086 4.7329392 1 1
  30.89362 35.173065 2 1
  .3012191  .4954418 1 2
 1.9271665  2.840809 1 2
  3.623339  5.476809 1 3
  17.95036 35.690914 3 3
 .05399648  .1044472 3 2
 10.553843 13.719444 1 3
  6.783232 13.273468 2 1
end
label values smoker3 smoker3
label def smoker3 1 "never", modify
label def smoker3 2 "former", modify
label def smoker3 3 "current", modify
label values cbmigroups3 cbmigroups3
label def cbmigroups3 1 "healthyweight", modify
label def cbmigroups3 2 "obese", modify
label def cbmigroups3 3 "overwight", modify
label def cbmigroups3 4 "underweight", modify


. sum mean_tda, detail

bone total dense area
-------------------------------------------------------------
Percentiles Smallest
1% .0353965 .0026717
5% .1441546 .0039107
10% .4248455 .0074923 Obs 1,029
25% 2.067551 .010919 Sum of Wgt. 1,029

50% 8.490792 Mean 14.15021
Largest Std. Dev. 16.82954
75% 20.56309 103.5658
90% 33.91737 104.3295 Variance 283.2335
95% 45.56728 140.7178 Skewness 2.552002
99% 77.17595 151.412 Kurtosis 13.92762


sum bd, detail

bone percentage density
-------------------------------------------------------------
Percentiles Smallest
1% .0176197 .0025312
5% .08834 .0026645
10% .2698087 .0047018 Obs 1,029
25% 1.631838 .0053402 Sum of Wgt. 1,029

50% 7.836907 Mean 12.44135
Largest Std. Dev. 13.79967
75% 19.00105 68.3148
90% 32.83851 69.05723 Variance 190.4309
95% 41.48195 71.48179 Skewness 1.528912
99% 56.84106 82.58966 Kurtosis 5.29763


Thanks,

Sonia