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Interpreting Regression Coefficients

by Karen Grace-Martin 32 Comments

Linear regression is one of the most popular statistical techniques.

Despite its popularity, interpretation of the regression coefficients of any but the simplest models is sometimes, well….difficult.

So let’s interpret the coefficients of a continuous and a categorical variable.  Although the example here is a linear regression model, the approach works for interpreting coefficients from any regression model without interactions, including logistic and proportional hazards models.

A linear regression model with two predictor variables can be expressed with the following equation:

Y = B0 + B1*X1 + B2*X2 + e.

The variables in the model are:

  • Y, the response variable;
  • X1, the first predictor variable;
  • X2, the second predictor variable; and
  • e, the residual error, which is an unmeasured variable.

The parameters in the model are:

  • B0, the Y-intercept;
  • B1, the first regression coefficient; and
  • B2, the second regression coefficient.

One example would be a model of the height of a shrub (Y) based on the amount of bacteria in the soil (X1) and whether the plant is located in partial or full sun (X2).

Height is measured in cm, bacteria is measured in thousand per ml of soil, and type of sun = 0 if the plant is in partial sun and type of sun = 1 if the plant is in full sun.

Let’s say it turned out that the regression equation was estimated as follows:

Y = 42 + 2.3*X1 + 11*X2

Interpreting the Intercept

B0, the Y-intercept, can be interpreted as the value you would predict for Y if both X1 = 0 and X2 = 0.

We would expect an average height of 42 cm for shrubs in partial sun with no bacteria in the soil. However, this is only a meaningful interpretation if it is reasonable that both X1 and X2 can be 0, and if the data set actually included values for X1 and X2 that were near 0.

If neither of these conditions are true, then B0 really has no meaningful interpretation. It just anchors the regression line in the right place. In our case, it is easy to see that X2 sometimes is 0, but if X1, our bacteria level, never comes close to 0, then our intercept has no real interpretation.

Interpreting Coefficients of Continuous Predictor Variables

Since X1 is a continuous variable, B1 represents the difference in the predicted value of Y for each one-unit difference in X1, if X2 remains constant.

This means that if X1 differed by one unit (and X2 did not differ) Y will differ by B1 units, on average.

In our example, shrubs with a 5000 bacteria count would, on average, be 2.3 cm taller than those with a 4000/ml bacteria count, which likewise would be about 2.3 cm taller than those with 3000/ml bacteria, as long as they were in the same type of sun.

(Don’t forget that since the bacteria count was measured in 1000 per ml of soil, 1000 bacteria represent one unit of X1).

Interpreting Coefficients of Categorical Predictor Variables

Similarly, B2 is interpreted as the difference in the predicted value in Y for each one-unit difference in X2 if X1 remains constant. However, since X2 is a categorical variable coded as 0 or 1, a one unit difference represents switching from one category to the other.

B2 is then the average difference in Y between the category for which X2 = 0 (the reference group) and the category for which X2 = 1 (the comparison group).

So compared to shrubs that were in partial sun, we would expect shrubs in full sun to be 11 cm taller, on average, at the same level of soil bacteria.

Interpreting Coefficients when Predictor Variables are Correlated

Don’t forget that each coefficient is influenced by the other variables in a regression model. Because predictor variables are nearly always associated, two or more variables may explain some of the same variation in Y.

Therefore, each coefficient does not measure the total effect on Y of its corresponding variable, as it would if it were the only variable in the model.

Rather, each coefficient represents the additional effect of adding that variable to the model, if the effects of all other variables in the model are already accounted for. (This is called Type 3 regression coefficients and is the usual way to calculate them. However, not all software uses Type 3 coefficients, so make sure you check your software manual so you know what you’re getting).

This means that each coefficient will change when other variables are added to or deleted from the model.

For a discussion of how to interpret the coefficients of models with interaction terms, see Interpreting Interactions in Regression.

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Interpreting Linear Regression Coefficients: A Walk Through Output
Learn the approach for understanding coefficients in that regression as we walk through output of a model that includes numerical and categorical predictors and an interaction.

Tagged With: categorical predictor, continuous predictor, Intercept, interpreting regression coefficients, linear regression

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Reader Interactions

Comments

  1. Ilana says

    November 11, 2020 at 8:34 am

    I have two binary independent variables how can I determine other then looking at the coefficient that one is stronger than the other? is there some test I need to do?

    Reply
  2. Jaume says

    May 5, 2020 at 11:22 am

    Absolutely clarifying, both this post and the one on interaction.

    Thank you very much.

    Reply
  3. Anna says

    January 14, 2020 at 3:56 am

    Hey Karen! Thanks for your explanation.
    What if I have a regression results table where race is coded as 1=black, 2= white and the coefficient for “race” is, for example, .13? How do I know how to interpret this? Is it possible to interpret this in magnitude? Thanks for your reply.

    Reply
    • Karen Grace-Martin says

      January 24, 2020 at 10:09 am

      Anna, you’d have to make sure that you’ve told your software that race is categorical. If you did, your software will dummy code it for you. If you can’t do that (depending on which software and which procedure you’re using) you’ll have to recode that variable into 1s and 0s.

      See this: https://www.theanalysisfactor.com/making-dummy-codes-easy-to-keep-track-of/

      Reply
  4. Mahdi says

    October 24, 2019 at 10:35 am

    Where can I get the dataset from (for this example)?

    Reply
  5. Paul says

    September 23, 2019 at 9:35 am

    Thanks for the excellent explanation. For clarity, I have a continuous dependent variable (annual change in quality of life score) and a binary independent variable (Control = 0, Treatment = 1), amongst other covariates. My coefficient is 1.3 (CI 0.41 to 2.19). Does this mean for each 1 point increase in Treatment group QoL score there is on average a 1.3 increase in control group? I am puzzled that the lower CI is 0.41. Would this mean that if the lower CI was true then there would be a 0.4 increase in control for each 1 point increase in treatment? Or is it that on average the QoL score is 0.4 higher for the control group? Many thanks

    Reply
  6. Manu Yaw says

    March 24, 2019 at 3:49 am

    How do I enter a categorical independent variable of 4 levels in stats.
    For example , marital status (single, married, divorced, separated)
    Thank you

    Reply
    • Karen Grace-Martin says

      April 11, 2019 at 9:41 am

      Hi Manu,

      The short answer is you need three Yes/No variables, each coded 1=yes and 0=no, for three of your four categories. It would take a while to walk you through this. We have a training on it in our membership program: https://www.theanalysisfactor.com/member-dummy-effect-coding/

      Reply
  7. Niru says

    March 20, 2019 at 9:11 pm

    Interesting read. I have a general question. Suppose we are comparing the coefficients of different models. Let’s say model 1 contains variables x1,x2,x3 and model two contains x1,x2,x3,x5. I do know that if there is a drastic difference in coefficients then there’s a potential multicollinearity problem. What if regardless of what’s in the model and what’s added, and the coefficients do not change. Does this simply imply there’s no multicollinearity?

    Reply
    • Karen Grace-Martin says

      March 21, 2019 at 3:27 pm

      Yes.

      Reply
  8. T says

    May 22, 2018 at 2:11 pm

    Thanks for this, terminology and notation are the most impenetrable parts of understanding statistics.

    Reply
  9. Liz says

    April 24, 2018 at 9:39 pm

    Hi,
    I have a dichotomous dependent variable and running a logitistic regression. The predictor of interest is a random effect of medical group. The dependent variable is quitter (Y/N) of smoking.
    1. How do I interpret the beta coefficient for medical group? For example, for medical group AX it is -.62.
    2. I want to adjust my percentage of quitters for medical group AX by -.62. Do I add this to the total number of quitters in AX or the percentage of quitters in AX or something else?

    Reply
    • Karen Grace-Martin says

      May 15, 2018 at 11:41 am

      Hi Liz,

      The beta coefficient in a logistic regression is difficult to interpret because it’s on a log-odds scale. I would suggest you start with this free webinar which explains in detail how to interpret odds ratios instead: Understanding Probability, Odds, and Odds Ratios in Logistic Regression

      Reply
  10. IB says

    March 10, 2018 at 11:14 am

    how do I interpret my intercept when my independent variable is gender and my dependent is continuous as it’s a big number and I don’t get it

    Reply
    • Karen Grace-Martin says

      March 4, 2019 at 11:55 am

      Hi IB,

      See this: https://www.theanalysisfactor.com/interpret-the-intercept/

      Reply
  11. anila says

    December 22, 2017 at 8:24 pm

    How to write the results of multiple regression analysis in our PhD thesis according to APA style? Can I have any example.

    Reply
    • Karen says

      January 29, 2018 at 12:23 pm

      Hi Anila, hmm. It’s been a while since I’ve had to use APA style.

      Reply
  12. Mark says

    April 4, 2017 at 12:00 pm

    How should I interpret the effects of an independent variable “age” (a continuous variable coded to range from (0) for the youngest to (1) for the oldest respondents) on my dependent variable “income” given a beta coefficient of 2.688823 ?

    Reply
  13. April says

    November 29, 2016 at 2:45 pm

    If you have a direction hypothesis for an IV, is it acceptable divide the two-tailed p-value for the t-value to obtain the one-tailed significance?

    Reply
  14. Juliet says

    October 16, 2016 at 1:09 am

    How do you interpret coefficients on discreet variables. For example, if sunlight was coded as 0 – no sunlight, 1 – partial sunlight and 2 – full sunlight, how would you interpret the coefficient on this independent variable?

    Reply
    • Jon says

      November 29, 2016 at 8:58 pm

      To handle categorical variables like in your example you would encode then into n-1 binary variables where n is the number of categories, see here for example: http://appliedpredictivemodeling.com/blog/2013/10/23/the-basics-of-encoding-categorical-data-for-predictive-models

      Reply
  15. Ahmed says

    August 23, 2016 at 12:43 am

    hello
    I used linear regression to control for IQ. How can I know if differences between two groups remain the same?

    Reply
  16. ENDALE Y says

    July 21, 2016 at 8:19 am

    Please make it easy and understandable.

    Reply
  17. Akosua says

    May 17, 2016 at 7:31 pm

    Please how do you interprete a regression result that show zero as the coefficient. Thank you

    Reply
    • Deniz says

      June 6, 2016 at 9:12 pm

      If B coefficient is 0 then, there is no relationship between dependent and independent variables.

      Reply
      • Kanu says

        October 5, 2016 at 1:04 pm

        Does this means that a B coefficient just over 0 lets say 0.58 isn’t as good as the one which is 1.11?

        What does the signs of the B coefficient’s means. Is it inverse association (-ve) and direct association (+ve) to the dependent variable?

        Reply
  18. Rupon Basumatary says

    November 7, 2015 at 11:53 am

    Makes easily understandable.

    Reply
  19. John says

    May 19, 2015 at 2:48 am

    In interpreting the coefficients of categorical predictor variables, what if X2 had several levels (several categories) instead of 0 and 1. Say, the soil was green, red, yellow or blue. How would you interpret quantitatively the differences in the coefficients? How much higher is the plant grown in green soil vs red soil?

    Reply
    • Gio says

      June 3, 2016 at 3:46 am

      John, you can always transform a multi level categorical variable in (levels-1) two level categorical variables.

      In your example the soil varaible would become:
      – Soil_green (1,0)
      – Soil_red (1,0)
      – Soil_Yellow (1,0)
      you do not need a Soil_Blue varaible because when all the above are 0 than you know it is a bout blue Soil

      Reply
      • him says

        July 17, 2016 at 7:58 pm

        FYI – The above is commonly referred to as “dummy coding”

        Reply
  20. Martins Ahmed says

    December 18, 2014 at 8:57 am

    Really appreciate this exposition. It has to a greater extent cleared some difficulties I have been experiencing when it comes to interpreting the results of coefficient of linear regression.

    Reply

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