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I like to learn how to interpret the effect estimates from a 4 level categorical predictor variable X , when my outcome Y is a continuous variable.

Y = outcome variable

X = 1,2,3,4 , categorical variable, where x=1 is my reference level.

These are the estimates from the model.

 Model :     Linear Model
 Intercept :  -0.363
 X (level2) : 0.406
 X (level3) : 0.135
 X (level4) : 0.2

Can I say that,

  • changing X from Level 1 (Reference level) to Level 2 will increase Y on an average by 0.406 units
  • changing X from Level 1 (Reference level) to Level 3 will increase Y on an average by 0.135 units

What will be the coefficient of my reference level X=1 ? is it 0 or 1.

Any help is much appreciated, Thanks.

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Can I say that

  • changing X from Level 1 (Reference level) to Level 2 will increase Y on an average by 0.406 units
  • changing X from Level 1 (Reference level) to Level 3 will increase Y on an average by 0.135 units

If you had no other predictors, yes. (Though your language here is causal which may be misleading if your data didn't come from a design that allows causal inferences. I'd perhaps rather say that the estimated difference in Y between Levels 1 and 2 is 0.406 etc.)

If you had other predictors you can say that the difference between levels 1 and 2 was .406 while adjusting for these other predictors.

The estimates you see in the output are not coefficients in the same sense as a coefficient for a continuous predictor. They are, as you have understood, levels' 2-4 differences from level 1. In a regression with a categorical predictor, you don't get an "overall" estimate for the categorical variable nor an overall estimate for each level. All estimates are comparisons between two levels.

However, the three estimates that you have are also the contrast estimates for level 1, just with reversed sign. E.g. if your reference level was 2, the estimate for level 1 would be -0.406.

(Also as you may notice, your intercept is the estimated mean of Y for level 1)

In a design like yours, it's customary to run post-hoc comparisons (e.g. in R using emmeans package and in SPSS through EMMEANS command). This way you get estimates of all pairwise differences between levels that you are interested in.

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  • $\begingroup$ @Sonitu, thank you very much Sonitu, one last clarification, since level1 is my reference can I say that the estimates for level 1 are treated as 0 ? 2) What if the outcome Y was log transformed ? how would this interpretation change ? Thanks. $\endgroup$
    – Science11
    Commented Oct 1, 2023 at 16:30
  • $\begingroup$ 1) Hm, there may be different ways to think about it but I'd say no. The estimates you get do not refer to any level but to differences/contrasts between two levels. And the value you compare levels 2-4 to is level 1's estimated mean. So the estimated mean would be the reference point. I suppose in some sense it could be seen representing a zero, but I don't think it's typical to describe the reference level like that. It's customary to just say that level 1 was the reference level. $\endgroup$
    – Sointu
    Commented Oct 2, 2023 at 6:32
  • $\begingroup$ 2) I personally rarely log-transform my variables so I'm a bit unsure, here is a good explanation and here are relevant responses from this forum. $\endgroup$
    – Sointu
    Commented Oct 2, 2023 at 6:34

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