Determinants of Plasma Carotene and Retinol

The following graphs and results have been created using STATA 5.0.

Introduction to Linear Models

Plasma Beta-carotene vs Age

. regress  beta_pl age

  Source |       SS       df       MS                  Number of obs =     315
---------+------------------------------               F(  1,   313) =    3.23
   Model |  107541.346     1  107541.346               Prob > F      =  0.0731
Residual |  10408097.0   313   33252.706               R-squared     =  0.0102
---------+------------------------------               Adj R-squared =  0.0071
   Total |  10515638.3   314   33489.294               Root MSE      =  182.35

------------------------------------------------------------------------------
 beta_pl |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
     age |   1.269719   .7060466      1.798   0.073      -.1194785    2.658917
   _cons |   126.2207    36.8661      3.424   0.001         53.684    198.7574
------------------------------------------------------------------------------

Plasma Retinol vs Alcohol

. regress  ret_pl alcohol

  Source |       SS       df       MS                  Number of obs =     315
---------+------------------------------               F(  1,   313) =    0.09
   Model |  4023.67732     1  4023.67732               Prob > F      =  0.7619
Residual |  13698094.5   313  43763.8802               R-squared     =  0.0003
---------+------------------------------               Adj R-squared = -0.0029
   Total |  13702118.2   314   43637.319               Root MSE      =  209.20

------------------------------------------------------------------------------
  ret_pl |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
 alcohol |   .2904922   .9580339      0.303   0.762      -1.594508    2.175493
   _cons |   601.8378    12.1985     49.337   0.000       577.8364    625.8393
------------------------------------------------------------------------------

Transformations

Plasma Beta-carotene vs Dietary Beta-carotene

log(Plasma Beta-carotene) vs log(Dietary Beta-carotene)

. regress logbeta logbdiet

  Source |       SS       df       MS           Number of obs =     314
---------+------------------------------        F(  1,   312) =   11.43
   Model |  6.06643516     1  6.06643516        Prob > F      =  0.0008
Residual |   165.58489   312    .5307208        R-squared     =  0.0353
---------+------------------------------        Adj R-squared =  0.0322
   Total |  171.651325   313  .548406788        Root MSE      =  .72851

------------------------------------------------------------------------
 logbeta |      Coef.   Std. Err.       t     P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------
logbdiet |   .2098945   .0620822      3.381   0.001 .0877417    .3320473
   _cons |   3.398304   .4663317      7.287   0.000 2.480751    4.315856
------------------------------------------------------------------------

Regression Diagnostics

Residual plot of Plasma Beta-carotene vs Dietary Beta-carotene

Residual plot of Log(Plasma Beta-carotene) vs Log (Dietary Beta-carotene)

Qualitative Predictor Variables

Categorical Variable (2 levels)

. regress ret_pl sex

  Source |       SS       df       MS                  Number of obs =     315
---------+------------------------------               F(  1,   313) =   10.99
   Model |   464927.21     1   464927.21               Prob > F      =  0.0010
Residual |  13237191.0   313  42291.3449               R-squared     =  0.0339
---------+------------------------------               Adj R-squared =  0.0308
   Total |  13702118.2   314   43637.319               Root MSE      =  205.65

------------------------------------------------------------------------------
  ret_pl |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
     sex |  -113.0165   34.08592     -3.316   0.001       -180.083   -45.94998
   _cons |   813.7546   64.67349     12.583   0.000       686.5048    941.0043
------------------------------------------------------------------------------

Categorical Variable (3 levels)

. regress ret_pl current former


  Source |       SS       df       MS                  Number of obs =     315
---------+------------------------------               F(  2,   312) =    3.79
   Model |  325058.873     2  162529.437               Prob > F      =  0.0236
Residual |  13377059.3   312  42875.1901               R-squared     =  0.0237
---------+------------------------------               Adj R-squared =  0.0175
   Total |  13702118.2   314   43637.319               Root MSE      =  207.06

------------------------------------------------------------------------------
  ret_pl |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
 current |  -20.23597   35.63969     -0.568   0.571       -90.3605    49.88857
  former |   60.93775   25.41492      2.398   0.017       10.93144    110.9441
   _cons |   583.3057   16.52545     35.297   0.000       550.7903    615.8211
------------------------------------------------------------------------------

Mixture of Continuous and Categorical Variables

. regress ret_pl female ret_diet

  Source |       SS       df       MS                  Number of obs =     314
---------+------------------------------               F(  2,   311) =    6.99
   Model |  589645.768     2  294822.884               Prob > F      =  0.0011
Residual |  13112355.6   311  42161.9151               R-squared     =  0.0430
---------+------------------------------               Adj R-squared =  0.0369
   Total |  13702001.4   313  43776.3622               Root MSE      =  205.33

------------------------------------------------------------------------------
  ret_pl |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
  female |  -119.3491   34.23946     -3.486   0.001      -186.7194   -51.97878
ret_diet |  -.0418489   .0243224     -1.721   0.086      -.0897063    .0060085
   _cons |   740.2494   39.13048     18.917   0.000       663.2555    817.2434
------------------------------------------------------------------------------

Two Qualitative Predictors

. regress ret_pl female current

  Source |       SS       df       MS                  Number of obs =     315
---------+------------------------------               F(  2,   312) =    6.61
   Model |  557326.746     2  278663.373               Prob > F      =  0.0015
Residual |  13144791.4   312  42130.7417               R-squared     =  0.0407
---------+------------------------------               Adj R-squared =  0.0345
   Total |  13702118.2   314   43637.319               Root MSE      =  205.26

------------------------------------------------------------------------------
  ret_pl |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
  female |  -114.7534   34.04135     -3.371   0.001      -181.7331    -47.7738
 current |  -49.91481   33.70498     -1.481   0.140      -116.2326    16.40299
   _cons |   709.0572   32.16627     22.043   0.000        645.767    772.3475
------------------------------------------------------------------------------

Two Qualitative and One Continuous Predictor

. regress ret_pl female current ret_diet

  Source |       SS       df       MS                  Number of obs =     315
---------+------------------------------               F(  3,   311) =    5.10
   Model |  643034.779     3  214344.926               Prob > F      =  0.0018
Residual |  13059083.4   311  41990.6218               R-squared     =  0.0469
---------+------------------------------               Adj R-squared =  0.0377
   Total |  13702118.2   314   43637.319               Root MSE      =  204.92

------------------------------------------------------------------------------
  ret_pl |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
  female |  -118.4176   34.08134     -3.475   0.001      -185.4768   -51.35848
 current |  -51.29626   33.66278     -1.524   0.129      -117.5318    14.93933
ret_diet |  -.0281254   .0196863     -1.429   0.154      -.0668605    .0106098
   _cons |   735.8418   37.18479     19.789   0.000       662.6763    809.0074
------------------------------------------------------------------------------

Analysis of Covariance

Plasma Retinol vs Age

. tab smoke,summarize(ret_pl)

            |  Summary of Plasma Retinol (ng/ml)
      smoke |        Mean   Std. Dev.       Freq.
------------+------------------------------------
    Current |   563.06977   206.57783          43
     Former |   644.24348   231.16762         115
      Never |   583.30573    187.6431         157
------------+------------------------------------
      Total |   602.79048   208.89547         315

. tab smoke,summarize(age)

            |            Summary of Age
      smoke |        Mean   Std. Dev.       Freq.
------------+------------------------------------
    Current |   44.534884   13.507227          43
     Former |   50.773913   13.972442         115
      Never |    51.22293   15.022461         157
------------+------------------------------------
      Total |   50.146032   14.575226         315

. anova ret_pl smoke age,cont(age)

                           Number of obs =     315     R-squared     =  0.0655
                           Root MSE      =  202.91     Adj R-squared =  0.0565

                  Source |  Partial SS    df       MS           F     Prob > F
              -----------+----------------------------------------------------
                   Model |  897521.329     3  299173.776       7.27     0.0001
                         |
                   smoke |  283599.164     2  141799.582       3.44     0.0332
                     age |  572462.455     1  572462.455      13.90     0.0002
                         |
                Residual |  12804596.8   311  41172.3371   
              -----------+----------------------------------------------------
                   Total |  13702118.2   314   43637.319   

. anova, regress detail

  Factor      Value          Value          Value          Value
  ----------------------------------------------------------------------
  smoke       1 1            2 2            3 3           

  Source |       SS       df       MS                  Number of obs =     315
---------+------------------------------               F(  3,   311) =    7.27
   Model |  897521.329     3  299173.776               Prob > F      =  0.0001
Residual |  12804596.8   311  41172.3371               R-squared     =  0.0655
---------+------------------------------               Adj R-squared =  0.0565
   Total |  13702118.2   314   43637.319               Root MSE      =  202.91

------------------------------------------------------------------------------
  ret_pl        Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
------------------------------------------------------------------------------
_cons        431.4374   43.82968      9.843   0.000       345.1972    517.6776
smoke
       1    -.4069026   35.32732     -0.012   0.991      -69.91768    69.10387
       2     62.26901   24.90767      2.500   0.013       13.26016    111.2779
       3    (dropped)
age          2.964851   .7951191      3.729   0.000       1.400358    4.529344
------------------------------------------------------------------------------

Two Factor Analysis of Variance

Plasma Retinol vs Smoking Status & Female

. tab smoke female,summarize(ret_pl)

   Means, Standard Deviations and Frequencies of Plasma Retinol (ng/ml)

      | female
     smoke |         0          1 |     Total
-----------+----------------------+----------
   Current | 598.85714  556.11111 | 563.06977
           | 289.61896  191.11265 | 206.57783
           |         7         36 |        43
-----------+----------------------+----------
    Former |     798.5  607.75269 | 644.24348
           |  323.1962  187.98373 | 231.16762
           |        22         93 |       115
-----------+----------------------+----------
     Never | 590.15385   582.6875 | 583.30573
           | 249.30799   182.1824 |  187.6431
           |        13        144 |       157
-----------+----------------------+----------
     Total |  700.7381  587.72161 | 602.79048
           | 307.80878  185.43069 | 208.89547
           |        42        273 |       315

. tab smoke female,summarize(age)

             Means, Standard Deviations and Frequencies of Age

           | female
     smoke |         0          1 |     Total
-----------+----------------------+----------
   Current | 55.142857  42.472222 | 44.534884
           | 13.885587  12.609489 | 13.507227
           |         7         36 |        43
-----------+----------------------+----------
    Former | 62.227273  48.064516 | 50.773913
           | 11.330245  13.184153 | 13.972442
           |        22         93 |       115
-----------+----------------------+----------
     Never | 60.615385     50.375 |  51.22293
           | 16.630987  14.636803 | 15.022461
           |        13        144 |       157
-----------+----------------------+----------
     Total | 60.547619  48.545788 | 50.146032
           | 13.469391  14.093135 | 14.575226
           |        42        273 |       315

. anova ret_pl smoke female

                           Number of obs =     315     R-squared     =  0.0523
                           Root MSE      = 204.337     Adj R-squared =  0.0432

                  Source |  Partial SS    df       MS           F     Prob > F
              -----------+----------------------------------------------------
                   Model |  716808.852     3  238936.284       5.72     0.0008
                         |
                   smoke |  251881.643     2  125940.821       3.02     0.0504
                  female |  391749.979     1  391749.979       9.38     0.0024
                         |
                Residual |  12985309.3   311  41753.4062   
              -----------+----------------------------------------------------
                   Total |  13702118.2   314   43637.319   

. anova, regress detail

  Factor      Value          Value          Value          Value
  ----------------------------------------------------------------------
  smoke       1 1            2 2            3 3           

  female      1 0            2 1           

  Source |       SS       df       MS                  Number of obs =     315
---------+------------------------------               F(  3,   311) =    5.72
   Model |  716808.852     3  238936.284               Prob > F      =  0.0008
Residual |  12985309.3   311  41753.4062               R-squared     =  0.0523
---------+------------------------------               Adj R-squared =  0.0432
   Total |  13702118.2   314   43637.319               Root MSE      =  204.34

------------------------------------------------------------------------------
  ret_pl        Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
------------------------------------------------------------------------------
_cons        574.6167   16.55271     34.714   0.000       542.0472    607.1861
smoke
       1    -28.62971   35.27696     -0.812   0.418       -98.0414    40.78198
       2     49.55186    25.3542      1.954   0.052      -.3356008    99.43932
       3    (dropped)
female
       1     104.9373   34.25874      3.063   0.002        37.5291    172.3456
       2    (dropped)
------------------------------------------------------------------------------

. anova ret_pl smoke female age,cont(age)

                           Number of obs =     315     R-squared     =  0.0785
                           Root MSE      =  201.82     Adj R-squared =  0.0666

                  Source |  Partial SS    df       MS           F     Prob > F
              -----------+----------------------------------------------------
                   Model |  1075395.83     4  268848.957       6.60     0.0000
                         |
                   smoke |  229636.971     2  114818.485       2.82     0.0612
                  female |  177874.501     1  177874.501       4.37     0.0375
                     age |  358586.977     1  358586.977       8.80     0.0032
                         |
                Residual |  12626722.3   310  40731.3624   
              -----------+----------------------------------------------------
                   Total |  13702118.2   314   43637.319   

. anova, regress detail

  Factor      Value          Value          Value          Value
  ----------------------------------------------------------------------
  smoke       1 1            2 2            3 3           

  female      1 0            2 1           

  Source |       SS       df       MS                  Number of obs =     315
---------+------------------------------               F(  4,   310) =    6.60
   Model |  1075395.83     4  268848.957               Prob > F      =  0.0000
Residual |  12626722.3   310  40731.3624               R-squared     =  0.0785
---------+------------------------------               Adj R-squared =  0.0666
   Total |  13702118.2   314   43637.319               Root MSE      =  201.82

------------------------------------------------------------------------------
  ret_pl        Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
------------------------------------------------------------------------------
_cons        451.4028   44.62898     10.115   0.000       363.5888    539.2169
smoke
       1     -9.73225   35.41985     -0.275   0.784      -79.42597    59.96147
       2     54.01194   25.08704      2.153   0.032       4.649529    103.3744
       3    (dropped)
female
       1     73.99278   35.40764      2.090   0.037       4.323086    143.6625
       2    (dropped)
age          2.455465   .8275627      2.967   0.003       .8271144    4.083815
------------------------------------------------------------------------------

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