Linear model: only the whole model significant?

Question

This is the result of a linear model (MatLab, LinearModel.fit):

Linear regression model:
        y ~ 1 + x1 + x2

    Estimated Coefficients:
                       Estimate     SE          tStat      pValue  
        (Intercept)     -0.24454     0.13033    -1.8763    0.081615
        x1             0.0032239    0.012141    0.26554     0.79446
        x2             0.0050906    0.012375    0.41136     0.68703


    Number of observations: 17, Error degrees of freedom: 14
    Root Mean Squared Error: 0.376
    R-squared: 0.529,  Adjusted R-Squared 0.462
    F-statistic vs. constant model: 7.86, p-value = 0.00515

The p-value for the model is significant, while there is no significance for the single predictors (nor for the intercept). How should this be interpreted?

Useful info: y is a behavioural variable (rating of visual images), while x1 and x2 are a brain activity parameter under 2 different conditions. Is a repeated measure design.

Model for x1 alone:

Linear regression model:
    y ~ 1 + x1

Estimated Coefficients:
                   Estimate     SE          tStat      pValue   
    (Intercept)     -0.21339      0.1031    -2.0698     0.056153
    x1             0.0081455    0.002008     4.0566    0.0010334


Number of observations: 17, Error degrees of freedom: 15
Root Mean Squared Error: 0.366
R-squared: 0.523,  Adjusted R-Squared 0.491
F-statistic vs. constant model: 16.5, p-value = 0.00103

Model for x2 alone:

Linear regression model:
    y ~ 1 + x2

Estimated Coefficients:
                   Estimate     SE           tStat      pValue    
    (Intercept)      -0.2619      0.10918    -2.3988      0.029895
    x2             0.0083287    0.0020395     4.0837    0.00097809


Number of observations: 17, Error degrees of freedom: 15
Root Mean Squared Error: 0.364
R-squared: 0.526,  Adjusted R-Squared 0.495
F-statistic vs. constant model: 16.7, p-value = 0.000978

Thanks.

Michael L. · Accepted Answer · 2016-11-29T09:27:54.287

4

I would strongly recommend you to check the correlation between $x_1$ and $x_2$ or you also could check whether $det(X'X)\approx0$. From my perspective this looks like the case of strong colinarity as the $R^2$ is high while the individual coefficients are not significant. In a non-theoretical and "handwaving" style I would say that OLS cannot decide to which covariate the effect should be attributed, as the are so similar. Note that this is only a problem for inference, but not for prediction as the OLSE $\hat{\beta}$ and $\hat{\sigma}^2$ remain unbiased.

How do the significance change if you leave $x_1$ or $x_2$ out?

edited Nov 29 '16 at 09:27

answered Nov 29 '16 at 09:20

Michael L.

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There is a very strong correlation (0.98). I was wondering if using a stepwise fitting...does it make sense to you? I updated the question to show how is the model with x1 / x2 alone. – smndpln Nov 29 '16 at 09:26
Can you describe to me what you mean be stepwise fitting? I would say that it does not make sense to estimate your model with both covariates if you want inference as they contain almost exactly the same information ($0.0081455\approx 0.0083287$). If you are estimating for the purpose of prediction you can leave both covariates in the model. – Michael L. Nov 29 '16 at 09:30
You're right. I'd say that the answer to the question is in your first reply. Is a result derived from strong colinearity. – smndpln Nov 29 '16 at 09:33

Linear model: only the whole model significant?

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