How can I interpret log transformed variables in terms of percent change in linear regression?



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How can I interpret log transformed variables in terms of percent change in linear regression?

Introduction

A typical use of a logarithmic transformation variable is to pull outlying data from a positively skewed distribution closer to the bulk of the data in a quest to have the variable be normally distributed. In regression analysis the logs of variables are routinely taken, not necessarily for achieving a normal distribution of the predictors and/or the dependent variable but for interpretability. Standard interpretation of coefficients in a regression analysis is that a one unit change in the independent variable results in the respective regression coefficient (beta) change in the expected value of the dependent variable while all the predictors are held constant. Interpreting a log transformed variable can be done in such a manner, however, such coefficients are routinely interpreted in terms of percent change (See Introductory Econometrics: A Modern Approach by Woolridge for discussion and derivation). Throughout this page we'll explore the interpretation in a simple linear regression setting with either the dependent variable, independent variable, or both variables are log-transformed. We'll use the hospital-level data from the Study on the Efficacy of Nosocomial Infection Control (data came from Applied Linear Regression Models 5th edition) where we'll explore the relationship between average length of stay (in days) for all patients in the hospital (length) and the average daily number of patients in the hospital (census). The focus of this page is model interpretation, not model logistics.

NOTE: The ensuing interpretation is applicable for only log base e (natural log) transformations.

clear 
input id length age risk culture xray beds msch region census nurses svcs
  1   7.13  55.7  4.1   9.0   39.6  279  2  4  207  241  60.0
  2   8.82  58.2  1.6   3.8   51.7   80  2  2   51   52  40.0
  3   8.34  56.9  2.7   8.1   74.0  107  2  3   82   54  20.0
  4   8.95  53.7  5.6  18.9  122.8  147  2  4   53  148  40.0
  5  11.20  56.5  5.7  34.5   88.9  180  2  1  134  151  40.0
  6   9.76  50.9  5.1  21.9   97.0  150  2  2  147  106  40.0
  7   9.68  57.8  4.6  16.7   79.0  186  2  3  151  129  40.0
  8  11.18  45.7  5.4  60.5   85.8  640  1  2  399  360  60.0
  9   8.67  48.2  4.3  24.4   90.8  182  2  3  130  118  40.0
 10   8.84  56.3  6.3  29.6   82.6   85  2  1   59   66  40.0
 11  11.07  53.2  4.9  28.5  122.0  768  1  1  591  656  80.0
 12   8.30  57.2  4.3   6.8   83.8  167  2  3  105   59  40.0
 13  12.78  56.8  7.7  46.0  116.9  322  1  1  252  349  57.1
 14   7.58  56.7  3.7  20.8   88.0   97  2  2   59   79  37.1
 15   9.00  56.3  4.2  14.6   76.4   72  2  3   61   38  17.1
 16  11.08  50.2  5.5  18.6   63.6  387  2  3  326  405  57.1
 17   8.28  48.1  4.5  26.0  101.8  108  2  4   84   73  37.1
 18  11.62  53.9  6.4  25.5   99.2  133  2  1  113  101  37.1
 19   9.06  52.8  4.2   6.9   75.9  134  2  2  103  125  37.1
 20   9.35  53.8  4.1  15.9   80.9  833  2  3  547  519  77.1
 21   7.53  42.0  4.2  23.1   98.9   95  2  4   47   49  17.1
 22  10.24  49.0  4.8  36.3  112.6  195  2  2  163  170  37.1
 23   9.78  52.3  5.0  17.6   95.9  270  1  1  240  198  57.1
 24   9.84  62.2  4.8  12.0   82.3  600  2  3  468  497  57.1
 25   9.20  52.2  4.0  17.5   71.1  298  1  4  244  236  57.1
 26   8.28  49.5  3.9  12.0  113.1  546  1  2  413  436  57.1
 27   9.31  47.2  4.5  30.2  101.3  170  2  1  124  173  37.1
 28   8.19  52.1  3.2  10.8   59.2  176  2  1  156   88  37.1
 29  11.65  54.5  4.4  18.6   96.1  248  2  1  217  189  37.1
 30   9.89  50.5  4.9  17.7  103.6  167  2  2  113  106  37.1
 31  11.03  49.9  5.0  19.7  102.1  318  2  1  270  335  57.1
 32   9.84  53.0  5.2  17.7   72.6  210  2  2  200  239  54.3
 33  11.77  54.1  5.3  17.3   56.0  196  2  1  164  165  34.3
 34  13.59  54.0  6.1  24.2  111.7  312  2  1  258  169  54.3
 35   9.74  54.4  6.3  11.4   76.1  221  2  2  170  172  54.3
 36  10.33  55.8  5.0  21.2  104.3  266  2  1  181  149  54.3
 37   9.97  58.2  2.8  16.5   76.5   90  2  2   69   42  34.3
 38   7.84  49.1  4.6   7.1   87.9   60  2  3   50   45  34.3
 39  10.47  53.2  4.1   5.7   69.1  196  2  2  168  153  54.3
 40   8.16  60.9  1.3   1.9   58.0   73  2  3   49   21  14.3
 41   8.48  51.1  3.7  12.1   92.8  166  2  3  145  118  34.3
 42  10.72  53.8  4.7  23.2   94.1  113  2  3   90  107  34.3
 43  11.20  45.0  3.0   7.0   78.9  130  2  3   95   56  34.3
 44  10.12  51.7  5.6  14.9   79.1  362  1  3  313  264  54.3
 45   8.37  50.7  5.5  15.1   84.8  115  2  2   96   88  34.3
 46  10.16  54.2  4.6   8.4   51.5  831  1  4  581  629  74.3
 47  19.56  59.9  6.5  17.2  113.7  306  2  1  273  172  51.4
 48  10.90  57.2  5.5  10.6   71.9  593  2  2  446  211  51.4
 49   7.67  51.7  1.8   2.5   40.4  106  2  3   93   35  11.4
 50   8.88  51.5  4.2  10.1   86.9  305  2  3  238  197  51.4
 51  11.48  57.6  5.6  20.3   82.0  252  2  1  207  251  51.4
 52   9.23  51.6  4.3  11.6   42.6  620  2  2  413  420  71.4
 53  11.41  61.1  7.6  16.6   97.9  535  2  3  330  273  51.4
 54  12.07  43.7  7.8  52.4  105.3  157  2  2  115   76  31.4
 55   8.63  54.0  3.1   8.4   56.2   76  2  1   39   44  31.4
 56  11.15  56.5  3.9   7.7   73.9  281  2  1  217  199  51.4
 57   7.14  59.0  3.7   2.6   75.8   70  2  4   37   35  31.4
 58   7.65  47.1  4.3  16.4   65.7  318  2  4  265  314  51.4
 59  10.73  50.6  3.9  19.3  101.0  445  1  2  374  345  51.4
 60  11.46  56.9  4.5  15.6   97.7  191  2  3  153  132  31.4
 61  10.42  58.0  3.4   8.0   59.0  119  2  1   67   64  31.4
 62  11.18  51.0  5.7  18.8   55.9  595  1  2  546  392  68.6
 63   7.93  64.1  5.4   7.5   98.1   68  2  4   42   49  28.6
 64   9.66  52.1  4.4   9.9   98.3   83  2  2   66   95  28.6
 65   7.78  45.5  5.0  20.9   71.6  489  2  3  391  329  48.6
 66   9.42  50.6  4.3  24.8   62.8  508  2  1  421  528  48.6
 67  10.02  49.5  4.4   8.3   93.0  265  2  2  191  202  48.6
 68   8.58  55.0  3.7   7.4   95.9  304  2  3  248  218  48.6
 69   9.61  52.4  4.5   6.9   87.2  487  2  3  404  220  48.6
 70   8.03  54.2  3.5  24.3   87.3   97  2  1   65   55  28.6
 71   7.39  51.0  4.2  14.6   88.4   72  2  2   38   67  28.6
 72   7.08  52.0  2.0  12.3   56.4   87  2  3   52   57  28.6
 73   9.53  51.5  5.2  15.0   65.7  298  2  3  241  193  48.6
 74  10.05  52.0  4.5  36.7   87.5  184  1  1  144  151  68.6
 75   8.45  38.8  3.4  12.9   85.0  235  2  2  143  124  48.6
 76   6.70  48.6  4.5  13.0   80.8   76  2  4   51   79  28.6
 77   8.90  49.7  2.9  12.7   86.9   52  2  1   37   35  28.6
 78  10.23  53.2  4.9   9.9   77.9  752  1  2  595  446  68.6
 79   8.88  55.8  4.4  14.1   76.8  237  2  2  165  182  48.6
 80  10.30  59.6  5.1  27.8   88.9  175  2  2  113   73  45.7
 81  10.79  44.2  2.9   2.6   56.6  461  1  2  320  196  65.7
 82   7.94  49.5  3.5   6.2   92.3  195  2  2  139  116  45.7
 83   7.63  52.1  5.5  11.6   61.1  197  2  4  109  110  45.7
 84   8.77  54.5  4.7   5.2   47.0  143  2  4   85   87  25.7
 85   8.09  56.9  1.7   7.6   56.9   92  2  3   61   61  45.7
 86   9.05  51.2  4.1  20.5   79.8  195  2  3  127  112  45.7
 87   7.91  52.8  2.9  11.9   79.5  477  2  3  349  188  65.7
 88  10.39  54.6  4.3  14.0   88.3  353  2  2  223  200  65.7
 89   9.36  54.1  4.8  18.3   90.6  165  2  1  127  158  45.7
 90  11.41  50.4  5.8  23.8   73.0  424  1  3  359  335  45.7
 91   8.86  51.3  2.9   9.5   87.5  100  2  3   65   53  25.7
 92   8.93  56.0  2.0   6.2   72.5   95  2  3   59   56  25.7
 93   8.92  53.9  1.3   2.2   79.5   56  2  2   40   14   5.7
 94   8.15  54.9  5.3  12.3   79.8   99  2  4   55   71  25.7
 95   9.77  50.2  5.3  15.7   89.7  154  2  2  123  148  25.7
 96   8.54  56.1  2.5  27.0   82.5   98  2  1   57   75  45.7
 97   8.66  52.8  3.8   6.8   69.5  246  2  3  178  177  45.7
 98  12.01  52.8  4.8  10.8   96.9  298  2  1  237  115  45.7
 99   7.95  51.8  2.3   4.6   54.9  163  2  3  128   93  42.9
100  10.15  51.9  6.2  16.4   59.2  568  1  3  452  371  62.9
101   9.76  53.2  2.6   6.9   80.1   64  2  4   47   55  22.9
102   9.89  45.2  4.3  11.8  108.7  190  2  1  141  112  42.9
103   7.14  57.6  2.7  13.1   92.6   92  2  4   40   50  22.9
104  13.95  65.9  6.6  15.6  133.5  356  2  1  308  182  62.9
105   9.44  52.5  4.5  10.9   58.5  297  2  3  230  263  42.9
106  10.80  63.9  2.9   1.6   57.4  130  2  3   69   62  22.9
107   7.14  51.7  1.4   4.1   45.7  115  2  3   90   19  22.9
108   8.02  55.0  2.1   3.8   46.5   91  2  2   44   32  22.9
109  11.80  53.8  5.7   9.1  116.9  571  1  2  441  469  62.9
110   9.50  49.3  5.8  42.0   70.9   98  2  3   68   46  22.9
111   7.70  56.9  4.4  12.2   67.9  129  2  4   85  136  62.9
112  17.94  56.2  5.9  26.4   91.8  835  1  1  791  407  62.9
113   9.41  59.5  3.1  20.6   91.7   29  2  3   20   22  22.9
end

We'll start of by looking at histograms of the length and census variable in its original metric and log-transformed state.

histogram length, bin(20)

gen l_length = log(length)
histogram l_length, bin(20)

histogram census, bin(20)

gen l_census = log(census)
histogram l_census, bin(10)

In both graphs, we saw how taking a log-transformation of the variable brought the outlying data points from the right tail towards the rest of the data. We'll start off by interpreting a linear regression model were the variables are in their original metric and then proceed to include the variables in their transformed state.

Variables in their Original Metric

For the first model with the variables in their original state, we'll regress average length of stay on the average daily number of patients in the hospital. Interpretation of the relationship is that a one person increase in the average daily number of patients in the hospital will change the average length of stay by 0.006 days.

regress length census

      Source |       SS       df       MS              Number of obs =     113
-------------+------------------------------           F(  1,   111) =   32.15
       Model |  91.8953338     1  91.8953338           Prob > F      =  0.0000
    Residual |  317.315045   111   2.8586941           R-squared     =  0.2246
-------------+------------------------------           Adj R-squared =  0.2176
       Total |  409.210379   112   3.6536641           Root MSE      =  1.6908

------------------------------------------------------------------------------
      length |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      census |   .0058911    .001039     5.67   0.000     .0038322      .00795
       _cons |   8.520931   .2546306    33.46   0.000     8.016363    9.025498
------------------------------------------------------------------------------

Dependent Variable Transformed

In this model, the dependent variable is in its log-transformed state and the independent variable is in its original metric. Comparing beta coefficient for census to the prior model we note that there is a big difference in coefficients, however we must recall the scale of the dependent variable changed states. In such models where the dependent variable has been log-transformed and the predictors are not, the format for interpretation is that dependent variable changes by 100*(beta coefficient) percent for a one unit increase in the independent variable while all other variable in the model are held constant. In this particular model we'd say that a increase of one person in the average daily number of patients in the hospital would results in a 0.06 percent change in the average length of stay.

regress l_length census

      Source |       SS       df       MS              Number of obs =     113
-------------+------------------------------           F(  1,   111) =   33.57
       Model |  .823665704     1  .823665704           Prob > F      =  0.0000
    Residual |  2.72378813   111  .024538632           R-squared     =  0.2322
-------------+------------------------------           Adj R-squared =  0.2253
       Total |  3.54745383   112  .031673695           Root MSE      =  .15665

------------------------------------------------------------------------------
    l_length |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      census |   .0005577   .0000963     5.79   0.000      .000367    .0007485
       _cons |   2.143387   .0235913    90.85   0.000     2.096639    2.190134
------------------------------------------------------------------------------

Independent Variable Transformed

In this model we are going to switch by having the dependent variable in its original metric and the independent variable being log-transformed. A comparison to the prior two models reveals that the regression coefficient is drastically different. Similar to the prior example the interpretation has a nice format, a one percent increase in the independent variable increases (or decreases) the dependent variable by (beta/100) units. In this particular model we'd say that a one percent increase in the average daily number of patients in the hospital would result in a (1.155/100) = 0.016 day increase in the average length of stay.

regress length l_census

      Source |       SS       df       MS              Number of obs =     113
-------------+------------------------------           F(  1,   111) =   34.90
       Model |  97.8749857     1  97.8749857           Prob > F      =  0.0000
    Residual |  311.335393   111  2.80482336           R-squared     =  0.2392
-------------+------------------------------           Adj R-squared =  0.2323
       Total |  409.210379   112   3.6536641           Root MSE      =  1.6748

------------------------------------------------------------------------------
      length |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    l_census |   1.155125   .1955447     5.91   0.000     .7676405     1.54261
       _cons |   3.937108   .9795713     4.02   0.000     1.996022    5.878194
------------------------------------------------------------------------------

Both Dependent and Independent Variables Transformed

Instances where both the dependent variables and independent variable(s) are log-transformed variables, the relationship is commonly referred in econometrics as elastic. In a regression setting, we'd interpret the elasticity as the percent change in y (the dependent variable), while x (the independent variable) increases by one percent. For this model we'd conclude that a one percent increase in the average daily number of patients in the hospital would yield a 0.11% increase in the average length of stay.

regress  l_length l_census

      Source |       SS       df       MS              Number of obs =     113
-------------+------------------------------           F(  1,   111) =   40.12
       Model |  .941798432     1  .941798432           Prob > F      =  0.0000
    Residual |   2.6056554   111  .023474373           R-squared     =  0.2655
-------------+------------------------------           Adj R-squared =  0.2589
       Total |  3.54745383   112  .031673695           Root MSE      =  .15321

------------------------------------------------------------------------------
    l_length |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    l_census |    .113311   .0178892     6.33   0.000     .0778625    .1487596
       _cons |   1.689884   .0896149    18.86   0.000     1.512306    1.867462
------------------------------------------------------------------------------

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