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Interpreting the Intercept in a Regression Model

by Karen Grace-Martin 42 Comments

Interpreting the Intercept in a regression model isn’t always a straightforward as it looks. Here’s the definition: the intercept (often labeled the constant) is the expected mean value of Y when all X=0. 

Start with a regression equation with one predictor, X.

If X sometimes equals 0, the intercept is simply the expected mean value of Y at that value. That’s meaningful.

If X never equals 0, then the intercept has no intrinsic meaning.  Both these scenarios are common in real data.In scientific research, the purpose of a regression model is to understand the relationship between predictors and the response.  If so, and if X never = 0, there is no interest in the intercept. It doesn’t tell you anything about the relationship between X and Y.

So whether the value of the intercept is meaningful or not, many times you’re just not interested in it. It’s not answering an actual research question.

You do need it to calculate predicted values, though.  In market research or data science, there is usually more interest in prediction, so the intercept is more important here.

When X never equals 0, but you want a meaningful intercept, simply consider centering X. If you re-scale X so that the mean or some other meaningful value = 0 (just subtract a constant from X), now the intercept has a meaning. It’s the mean value of Y at the chosen value of X.

If you have dummy variables in  your model, though, the intercept has more meaning.  Dummy coded variables have values of 0 for the reference group and 1 for the comparison group. Since the intercept is the expected mean value when X=0, it is the mean value only for the reference group (when all other X=0).

This is especially important to consider when the dummy coded predictor is included in an interaction term.  Say for example that X1 is a continuous variable centered at its mean.  X2 is a dummy coded predictor, and the model contains an interaction term for X1*X2.

The B value for the intercept is the mean value of X1 only for the reference group.  The mean value of X1 for the comparison group is the intercept plus the coefficient for X2.

It’s hard to give an example because it really depends on how X1 and X2 are coded. So I put together 6 situations in this follow up:

How to Interpret the Intercept in 6 Linear Regression Examples

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: centering, Interpreting intercept, interpreting regression coefficients, regression models

Related Posts

  • Interpreting (Even Tricky) Regression Coefficients – A Quiz
  • Interpreting Regression Coefficients in Models other than Ordinary Linear Regression
  • When NOT to Center a Predictor Variable in Regression
  • Centering and Standardizing Predictors

Reader Interactions

Comments

  1. ELsaadani says

    October 18, 2019 at 7:23 am

    Hello, i have a question for my test that i hope i will find the answer here.

    I made a calibration curve for my toxine with 6 differents concentration (in ng/mL) and i got a R2= 0.99 and y=10.273x – 20.395.
    Is it logic that the accuracy of intercept is higher than that of the slope ? and why this happened!

    Reply
  2. Alina says

    September 10, 2019 at 11:24 am

    Thanks so much for your explanations, Karen! I have a question: can I interpret the intercept (Y) in a regression model where my intercept is significant and two other predictors ( say X and Z), while X can never be zero but Z can be 0 ?
    In my case Y is a change score. If the intercept is not equal to zero and significant can I infer from this that there is an overall change?

    Reply
    • Karen Grace-Martin says

      October 28, 2019 at 10:14 am

      Hi Alina,

      The intercept is only interpretable if all predictors (X and Z) can be zero.

      Reply
  3. Iva says

    September 9, 2019 at 1:43 am

    Hi, I m analyzing logistic regression for my independent and dependent variables, form the regression coefficient I want to calculate risk score of the independent variables on dependent variable. but in the regression model i got few variables have significant association and others have no significant relation with my dependent variables. so when calculating my score should I consider the intercept of the model with all significant and non significant independent variables or I should analyse another logistic regression with only the independent variables those are significant and then should take that intercept value ? please guide me….

    Reply
  4. John says

    August 12, 2019 at 10:59 am

    thanks
    this is useful

    Reply
  5. Anika Islam lopa says

    March 19, 2019 at 9:12 am

    Hi, pls answer me,
    Can intercept be zero In regression analysis??

    Reply
    • Karen Grace-Martin says

      March 21, 2019 at 4:12 pm

      Sure. You just don’t want to force it to be.

      Reply
      • Mariam Amin says

        June 21, 2020 at 11:05 pm

        Thank you Karen for this answer

        Reply
  6. John says

    November 30, 2018 at 3:24 am

    Hi Karen,

    I’m using my model to calculate predicted values so I need to include the constant. I’m concerned however that whilst my 3 regression variables are significant, the constant is not. I’m concerned that the constant being not significant means that I can’t be confident about the predicted values.

    Thoughts?

    Thanks
    John

    Reply
    • Karen Grace-Martin says

      November 30, 2018 at 11:49 am

      Hi John,

      The p-value for the constant isn’t important. It’s testing the null hypothesis that the constant = 0.

      So even if the constant isn’t significantly different from 0, including it will still give you more accurate predicted values AND more accurate slopes than if you eliminate it. When you eliminate it, you set it to 0.

      Reply
  7. Eusebio Bonelli says

    January 17, 2018 at 4:18 am

    Everything is very open with a clear explanation of the issues.
    It was really informative. Your site is very helpful.
    Many thanks for sharing!

    Reply
  8. Tim says

    March 21, 2017 at 10:56 am

    Thank you for this. May I suggest that it may be really helpful to use an example with real data to help explain how this works? For me (and I expect for others too), it would make it much easier to understand.

    Reply
    • Bob says

      September 13, 2018 at 8:19 pm

      I 100% agree. YES

      Reply
  9. Samra Bukhari says

    October 25, 2016 at 5:36 am

    Why value of intercept is zero?
    If intercept is not in the model than what happened?

    Reply
  10. sk saha says

    July 26, 2016 at 3:04 am

    Is this is possible??the intercept term of a regression model is negative??Can u make me clear with an easy example plz???

    Reply
  11. Anil Bandyopadhyay says

    July 24, 2016 at 4:02 am

    Case Summaries
    Imp
    Type N Mean
    1 7 5.86
    2 4 5.75
    3 89 4.61
    Total 100 4.74
    How to interpret the above? 1 = Manufacturer; 2 = Distributor; 3 = Retailer of a product.

    RECODE Type (1=0) (2=0) (3=1) INTO retail.
    EXECUTE.
    REGRESSION
    /MISSING LISTWISE
    /STATISTICS COEFF OUTS R ANOVA
    /CRITERIA=PIN(.05) POUT(.10)
    /NOORIGIN
    /DEPENDENT Imp
    /METHOD=ENTER Size retail.

    How to interpret the above? 1 = Manufacturer; 2 = Distributor; 3 = Retailer of a product
    Anil Bandyopadhyay

    Reply
  12. shsdi says

    August 13, 2015 at 4:26 pm

    Can we use negstive intercept ?
    I have two nagative intercept what can i do

    Reply
    • Karen says

      August 20, 2015 at 5:29 pm

      Yes, intercepts can be negative even if Y can’t. This usually occurs when none of the X values are close to 0.

      Reply
  13. Sweety says

    May 19, 2015 at 3:23 am

    what if coefficient of regression is 1.797?

    Reply
  14. shahzad pirzada says

    May 13, 2015 at 8:51 am

    why intercept used negative any way??????????

    Reply
  15. chhaya khillare says

    March 25, 2015 at 4:48 am

    what happend if the intersept in not in minus. i m confuse i found some are in positive and some are in negative intesept..
    please clarify.

    Reply
  16. Ahmed says

    January 9, 2015 at 1:53 pm

    Hi,
    What if you use a tobit model where the dependent variable takes values of zero or more than zero and you get a negative intercept. You run the tobit model and you observe a negative constant. What does this mean in this case?

    Reply
  17. midhat says

    January 2, 2015 at 4:46 am

    What is the fixed and estimated value in regression equation? a or b?
    Please reply asap .☺

    Reply
  18. Phillip says

    December 16, 2014 at 1:40 pm

    Wow. Thanks so helpful

    Reply
  19. T J says

    November 10, 2014 at 10:19 am

    Does the value of the intercept ever change, for example, when you are trying to interpret significant interaction terms? Say, I have three predictors, X1, X2, and X3 in a significant interaction. X1 is a four level categorical, and the other two are centered and continuous. So, the intercept can be defined as one level of X1, X2 = mean, and X3 = mean. And I can see that there is a 3-way interaction in which for one level of X1 (relative to the intercept as defined above), as X2 goes up and X3 goes up by one unit, I need to adjust the estimate for the simple effect of that level of X1 by some amount. But I am confused how this then relates back to the intercept. Is it really still defined as above? Or once I start considering the interaction, do I also change the designation for X2 and X3 in the intercept? Thanks.

    Reply
  20. Taylor Nelson says

    September 2, 2014 at 9:02 pm

    What if you intercept isn’t significant, and you are using a dummy variable? Should you still use it in your prediction equation?

    Thanks!

    Reply
    • Karen says

      September 3, 2014 at 9:44 am

      Yes. 🙂

      Reply
      • Usha Nepal says

        November 26, 2019 at 12:31 am

        I have regression equation y=74.626+1.2x, then how can the meaning of y-intercept be interpreted?

        Reply
  21. OYETUNDE ADEMOLA ISMAIL says

    August 22, 2014 at 11:17 am

    I will like to ask, when dealing with two indepents variable and our priori expectation for our coefficient is said to be greater than zero, what of if it happens that the intercept is negative, are we saying this is significant or insignificant ?

    Reply
  22. Joe says

    April 21, 2014 at 9:16 am

    If I built 3 index variables and several dummy variables, then was told to test to see if there is a relationship between how satisfied employees are in their job (index variable 1) and how they see the environment around them. I ran the regression and my results were that my two Independent Variable Indexes were significant, but my constant was not. The Adj. R-square was .608 and the F Sig was .000.

    What am I doing wrong?? Or what can I interpret from my results?

    Reply
  23. John says

    March 11, 2014 at 9:52 pm

    In a negative binomial regression, what would it mean if the Exp(B) value for the intercept falls below the lower limit of the 95% Confidence Interval?

    Reply
    • Karen says

      April 4, 2014 at 9:57 am

      Hmm, not sure I understand your question. CI for what?

      Reply
  24. Irena says

    December 4, 2013 at 12:55 pm

    Hi! What happens if all of my variables can be 0 which had a significant regressions coefficient? (I have four Xs, 3 of them have a significant coefficient and can be 0 as they are either dummies or are on a scale from 0 and there are 0s in the sample, but one of the Xs cannot be 0. It’s also the one with not significant coefficient.)
    Thanks!

    Reply
    • Karen says

      December 9, 2013 at 10:50 am

      Hi Irena,

      If ANY of the Xs can’t be 0, then the intercept doesn’t mean anything. Or rather, it’s just an anchor point, but it’s not directly interpretable.

      Reply
  25. kamil says

    November 25, 2013 at 5:20 pm

    als would like to as about, if we decrease sample by half will SSE, SSR, SST increase or decrease, a bit confused.

    Reply
    • Karen says

      November 25, 2013 at 5:58 pm

      None would change, theoretically. Sums of Squares are not directly affected by sample size.

      Reply
  26. kamil says

    November 25, 2013 at 5:17 pm

    does this mean that if education is =to zero, i.e no education, then the expected mean of y =-5

    Reply
    • Karen says

      November 25, 2013 at 5:57 pm

      Yes.

      Reply
  27. kamil says

    November 23, 2013 at 10:28 am

    quetion: if wage =-5+10*years of education and wage is measure in 1000s; how do you interpret the coeffficient and does the intercept make sende

    Reply
    • Karen says

      November 25, 2013 at 3:29 pm

      This sounds like a homework question, so I’m going to try to answer only by getting you to think through it.

      Since the intercept ALWAYS is the mean of Y (1000 of dollars or whatever the currency is) when X=0, it will only be meaningful if it’s meaningful that X=0 AND if there are examples in the data set. Is there anyone in the data set with years of education = 0?

      Reply
  28. silvinamontes@yahoo.es says

    February 2, 2013 at 6:25 pm

    I’d like to now why the need for a column of ones in the model to account for the intercept. I would need a basic answer, since I’m not a mathematician. Thank you.

    Reply
    • Karen says

      March 4, 2013 at 11:08 am

      In the X matrix, each column is the value of the X that is multiplied by that regression coefficient.

      Since the intercept isn’t multiplied by any values of X, we put in 1s.

      It makes all the matrix algebra work out.

      Karen

      Reply

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