# Linear model: only the whole model significant?

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
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
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
F-statistic vs. constant model: 16.7, p-value = 0.000978


Thanks.

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?
• 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 9:30