Introduction

In this page, we will discuss how to interpret a regression model when some
variables in the model have been log transformed.
The example data can be downloaded here (the file is in .csv format). The
variables in the data set are writing, reading, and math scores ( (textbf{write}), (textbf{read}) and (textbf{math})),
the log transformed writing (lgwrite) and log transformed math scores (lgmath)
and (textbf{female}). For these examples, we have taken the natural log (ln).  All the examples are done in Stata, but they can be easily
generated in any statistical package. In the examples below, the variable ( textbf{write} ) or its log
transformed version will be used as the outcome variable. The examples are used for
illustrative purposes and are not intended to make substantive sense. Here is a
table of different types of means for variable ( textbf{write} ).

    Variable |    Type        Obs        Mean       [95% Conf. Interval]
-------------+----------------------------------------------------------
       write | Arithmetic     200      52.775        51.45332   54.09668 
             |  Geometric     200     51.8496        50.46854   53.26845 
             |   Harmonic     200    50.84403        49.40262   52.37208 
------------------------------------------------------------------------

Outcome variable is log transformed

Very often, a linear relationship is hypothesized between a log transformed
outcome variable and a group of predictor variables. Written mathematically, the
relationship follows the equation

begin{equation}
log(y_i) = beta_0 + beta_1 x_{1i} + cdots + beta_k x_{ki} + e_i ,
end{equation}

where (y) is the outcome variable and (x_1, cdots, x_k) are the predictor variables.
In other words, we assume that (log(y) – mathbf{x}^T boldsymbolbeta )
is normally distributed, (or (y) is log-normal conditional on all the covariates). Since this is just
an ordinary least squares regression, we can easily interpret a
regression coefficient, say (beta_1 ), as the expected change in log of ( y) with respect to a one-unit
increase in (x_1) holding all other variables at any fixed value, assuming that (x_1)
enters the model only as a main effect. But what if we want to know
what happens to the outcome variable (y) itself for a one-unit increase in (x_1)?
The natural way to do this is to interpret the exponentiated regression
coefficients, ( exp(beta)), since exponentiation is
the inverse of logarithm function.

Let’s start with the intercept-only model.

------------------------------------------------------------------------------
     lgwrite |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
   intercept |   3.948347   .0136905   288.40   0.000      3.92135    3.975344
------------------------------------------------------------------------------

(log(textbf{write}) = beta_0 = 3.95).

We can say that (3.95) is the unconditional expected mean of log of ( textbf{write} ).
Therefore the exponentiated value is (exp(3.948347) = 51.85). This is the geometric mean of
( textbf{write} ). The emphasis here is that it is the geometric mean
instead of the arithmetic mean. OLS regression of the original variable (y) is used to
to estimate the expected arithmetic mean and OLS regression of the log
transformed outcome variable is to estimated the expected geometric mean of the
original variable.

Now let’s move on to a model with a single binary predictor variable.

------------------------------------------------------------------------------
     lgwrite |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      female |   .1032614   .0265669     3.89   0.000      .050871    .1556518
   intercept |    3.89207   .0196128   198.45   0.000     3.853393    3.930747
------------------------------------------------------------------------------

( begin{split}
log(textbf{write}) & = beta_0 + beta_1 times textbf{female} \
& = 3.89 + .10 times textbf{female} .
end{split} )

Before diving into the interpretation of these parameters, let’s get the
means of our dependent variable, ( textbf{write} ), by gender.

males

    Variable |    Type        Obs        Mean       [95% Conf. Interval]
-------------+----------------------------------------------------------
       write | Arithmetic      91    50.12088        47.97473   52.26703 
             |  Geometric      91    49.01222         46.8497   51.27457 
             |   Harmonic      91    47.85388         45.6903   50.23255 
------------------------------------------------------------------------

females

    Variable |    Type        Obs        Mean       [95% Conf. Interval]
-------------+----------------------------------------------------------
       write | Arithmetic     109    54.99083        53.44658   56.53507 
             |  Geometric     109    54.34383        52.73513    56.0016 
             |   Harmonic     109    53.64236        51.96389   55.43289 
------------------------------------------------------------------------

Now we can map the parameter estimates to the geometric means for the two
groups. The intercept of (3.89) is the log of geometric mean of ( textbf{write} ) when
(textbf{female} = 0), i.e., for males. Therefore, the exponentiated value of it is the
geometric mean for the male group: (exp(3.892) = 49.01). What can we say about the
coefficient for (textbf{female})? In the log scale, it is the difference in
the expected arithmetic means of the log of (textbf{write}) between the female
students and male students. In the original scale of the variable (textbf{write}),
it is the ratio of the geometric mean of (textbf{write}) for female students over
the geometric mean of (textbf{write}) for male students, (exp(.1032614) =
54.34383 / 49.01222 = 1.11). In terms of percent change, we can say that switching
from male students to female students, we expect to see about (11%) increase in
the geometric mean of writing scores.

Last, let’s look at a model with multiple predictor variables.

------------------------------------------------------------------------------
     lgwrite |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      female |    .114718   .0195341     5.87   0.000      .076194     .153242
        read |   .0066305   .0012689     5.23   0.000     .0041281    .0091329
        math |   .0076792   .0013873     5.54   0.000     .0049432    .0104152
   intercept |   3.135243   .0598109    52.42   0.000     3.017287    3.253198
------------------------------------------------------------------------------

( begin{split}
log(textbf{write}) & = beta_0 + beta_1 times textbf{female} + beta_2 times textbf{read} + beta_3 times textbf{math} \
& = 3.135 + .115 times textbf{female} + .0066 times textbf{read} + .0077 times textbf{math} .
end{split} )

The exponentiated coefficient ( exp(beta_1) )
for (textbf{female}) is the ratio of the expected geometric mean for the female
students group over the expected geometric mean for the male students group, when
(textbf{read}) and (textbf{math}) are held at some fixed value. Of course, the
expected geometric means for the male and female students group will be
different for different values of (textbf{read}) and (textbf{math}). However, their ratio is a
constant: ( exp(beta_1) ). In our
example, ( exp(beta_1) = exp(.114718) approx 1.12 )
. We can say that writing scores will be (12%) higher for the female
students than for the male students. For the variable (textbf{read}), we
can say that for a one-unit increase in (textbf{read}), we expect to see about a (0.7%)
increase in writing score, since (exp(.0066305) = 1.006653 approx 1.007).
For a ten-unit increase in (textbf{read}), we expect to see about a (6.9%) increase in writing score, since ( exp(.0066305 times 10) = 1.0685526 approx 1.069).

The intercept becomes less interesting when the predictor variables are not centered and are continuous. In this particular model, the intercept is the expected
mean for (log(textbf{write}) ) for male ( ( textbf{female} = 0 ) ) when (textbf{read}) and (textbf{math}) are equal to zero.

In summary, when the outcome variable is log transformed, it is natural to
interpret the exponentiated regression coefficients.  These values correspond to changes
in the ratio of the expected geometric means of the original outcome variable.

Some (not all) predictor variables are log transformed

Occasionally, we also have some predictor variables being log transformed. In
this section, we will take a look at an example where some predictor variables
are log-transformed, but the outcome variable is in its original scale.

------------------------------------------------------------------------------
       write |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      female |   5.388777   .9307948     5.79   0.000     3.553118    7.224436
      lgmath |   20.94097   3.430907     6.10   0.000     14.17473     27.7072
      lgread |   16.85218   3.063376     5.50   0.000     10.81076    22.89359
   intercept |  -99.16397   10.80406    -9.18   0.000    -120.4711   -77.85685
------------------------------------------------------------------------------

Written in equation, we have

( begin{split}
textbf{write} & = beta_0 + beta_1 times textbf{female} + beta_2 times log(textbf{math}) + beta_3 times log(textbf{read}) \
& = -99.164 + 5.389 times textbf{female} + 20.941 times log(textbf{math}) + 16.852 times log(textbf{read}).
end{split} )

Since this is an OLS regression, the interpretation of the regression
coefficients for the non-transformed variables are unchanged from an OLS
regression without any transformed variables. For example, the
expected mean difference in writing scores between the female and male students is about (5.4) points, holding
the other predictor variables constant. On the other hand,
due to the log transformation, the estimated effects of ( textbf{math} ) and ( textbf{read} ) are no
longer linear, even though the effect of ( log(textbf{math}) ) and ( log(textbf{read}) ) are
linear. The plot below shows the curve of predicted values against the reading
scores for the female students group holding math score constant.

How do we interpret the coefficient of (16.852 ) for the variable of log of
reading score? Let’s take two values of reading score, ( r_1 ) and ( r_2 ). The expected
mean difference in writing score at ( r_1 ) and ( r_2 ), holding the other predictor variables constant, is
( textbf{write}(r_2) – textbf{write}(r_1) = beta_3 times [ log(r_2) – log(r_1) ] = beta_3 times [log(r_2 / r_1)] ).
This means that as long as the percent increase in ( textbf{read} )
(the predictor variable) is fixed, we will see the same difference in writing
score, regardless where the baseline reading score is. For example, we can say
that for a (10%) increase in reading score, the difference in the expected mean
writing scores will be always (beta_3 times log(1.10) = 16.85218 times log(1.1) approx 1.61 ).

Note:

Recalling the Taylor expansion of the function (f(x) = log(1 + x) ) around (x_0 = 0), we have ( log(1 + x) = x + mathcal{O}(x^2)).
Therefore, for a small change in the predictor variable we can approximate the difference in the expected mean of the dependent variable
by multiplying the coefficient by the change in the predictor variable. In our example we can say that for a (1%) increase in reading score,
the difference in the expected mean writing scores will be approximately (beta_3 times 0.01 = 16.85218 times 0.01 = .1685218).
If we use the log, the exact value will be (beta_3 times log(1.01) = 16.85218 times log(1.01) = .1676848 ).

Both the outcome variable and some predictor variables are log
transformed

What happens when both the outcome variable and predictor variables are log
transformed? We can combine the two previously described situations into one.
Here is an example of such a model.

------------------------------------------------------------------------------
     lgwrite |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      female |   .1142399   .0194712     5.87   0.000       .07584    .1526399
      lgmath |   .4085369   .0720791     5.67   0.000     .2663866    .5506872
        read |   .0066086   .0012561     5.26   0.000     .0041313    .0090859
   intercept |   1.928101   .2469391     7.81   0.000     1.441102    2.415099
------------------------------------------------------------------------------

Written as an equation, we can describe the model:

( begin{split}
log(textbf{write}) & = beta_0 + beta_1 times textbf{female} + beta_2 times log(textbf{math}) + beta_3 times textbf{read} \
& = 1.928101 + .1142399 times textbf{female} + .4085369 times log(textbf{math}) + .0066086 times textbf{read}.
end{split} )

For variables that are not transformed, such as (textbf{female}), its
exponentiated coefficient is the ratio of the geometric
mean for the female to the geometric mean for the male students group. For
example, in our example, we can say that the expected percent increase in
geometric mean from male student group to female student group is about (12% )
holding other variables constant, since (exp(.1142399) approx 1.12 ). For reading score,
we can say that for a one-unit increase in reading score, we expected to see
about (0.7%) of increase in the geometric mean of writing score, since (exp(.0066086)
= 1.007).

Now, let’s focus on the effect of (textbf{math}). Take two values of (textbf{math}), (m_1)
and (m_2), and hold the other predictor variables at any fixed value. The equation above yields

(
begin{equation}
log(textbf{write}(m_2)) – log(textbf{write}(m_1)) = beta_2 times [log(m_2) – log(m_1)]
end{equation} )

It can be simplified to ( log[textbf{write}(m_2)/textbf{write}(m_1)) = beta_2 times[log(m_2/m_1)] ), leading to

(
begin{equation}
dfrac{textbf{write}(m_2)}{textbf{write}(m_1)} = left(dfrac{m_2}{m_1} right) ^ {beta_2}
end{equation} )

This tells us that as long as the ratio of the
two math scores, (m_2/m_1) stays the same, the expected ratio of the outcome variable,
(textbf{write}), stays the same. For example, we can say that for any (10%) increase
in (textbf{math}) score, the expected ratio of the writing
score will be ( (1.10) ^ {beta_2} = (1.10) ^ .4085369 = 1.0397057 ). In other
words, we expect about (4%) increase in writing score when math score increases by
(10%).

Note:

Here also we can use an approximation method. Since, ( (1 + x) ^ a approx 1 + ax) for a small value of ( |a|x ),
therefore for a small change in the predictor variable we can approximate the expected ratio of the of the dependent variable
by multiplying the coefficient by the ratio of the change in the predictor variable. For example, we can say that for any (1%) increase
in (textbf{math}) score, the expected ratio of the writing
score is approximately ( 1 + .01 times {beta_2} = 1 + .01 times .4085369 = 1.004085 ).
The exact value will be ( (1.01) ^ {beta_2} = (1.01) ^.4085369 = 1.004073 ).