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Hi I am having trouble acquiring the final results for presentation. The results from a multiple regression are different to my results in a simple linear regression.

For example, the multiple regression model mul <- lm(Response ~ Temp + Wave + Overcast, data = env) gives me a p-value for 'Temp' of $0.01$, while the others are $>0.05$.

However, a simple linear regression sim <- lm(Response ~ Temp, data =env) returns a p-value for 'Temp' of $0.03$.

Which result should I be presenting, and also how to I interpret the relationship between the predictors in the multiple regression?

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  • $\begingroup$ Hint: look at the degrees of freedom for the t statistics in the simple and multiple linear regression. Then compare them. $\endgroup$ – Stat Jan 31 '16 at 1:30
  • $\begingroup$ There is a superb answer to this general question here. In the terminology of that answer, think of each linear regression coefficient as the result of ignoring all of the other variables, while each multiple regression coefficient is controlling for all the other variables. $\endgroup$ – EdM Jan 31 '16 at 15:23
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You are comparing two entirely different models. Which one you choose has a lot to do with experience in the field of study, aided by things such as adjusted R squares and ANOVA testing of both models side-by-side.

In multiple regression, the idea is that when we add a regressor ($x_2$) to a pre-existing model with just one regressor ($x_1$), we are regressing the "dependent variable" $y$ not over $x_2$, but over the residuals of the regression of $x_2$ over $x_1$. This changes everything.

If we set up the model as:

$y = \color{blue}{5}\, x_2 + \color{red}{15}\, x_1$ with both $x_1$ and $x_2$ generated as random $\sim N(0,1)$, the correlation $y$ with $x_1$ will be much higher than with $x_2$:

The regression of $y$ over $x_2$ in isolation will not capture the slope we set p of $5$, because $x_2$ will be compensating for the absence of the main explanatory variable in the model, i.e. $x_1$. The slope of $x_2$ will be, in fact, very close to the coefficient we assigned to $x_1$: $\color{red}{15}$. Leaving the intercept out, summary(lm(y ~ x2 - 1))$coefficients will return a slope of $\color{red}{15.42}$.

However, if we now include $x_1$ in the regression in a sneaky way, and instead of calling lm(y ~ x1 + x2 - 1), we first regress $x_2$ over $x_1$ and keep the residuals before "tossing" $x_1$ as errors <- residuals(lm(x2 ~ x1 - 1)) and then call summary(lm(y ~ errors - 1))$coefficients the slope will be $\color{blue}{4.681616}$, very close to the coefficient we set up for $x_2$, and... here comes the punch of the story... identical to coefficient for $x_2$ in summary(lm(y ~ x2 + x1 - 1))$coefficients: $x_2 \,\,\color{blue}{4.681616}$. The coefficient for $x_1$ will be $x_1 \,\,\color{red}{15.091305}$.

So ignoring the hidden confounder $x_1$ in the model $y \sim x_2$ forced $x_2$ to explain all by itself as much of the variation in $y$ as possible, resulting in a completely different slope as compared to the more accurate $y \sim x_1 + x_2$. You can check the concept of omitted variable bias.

It makes sense that the $p$-values are going to be significant in both instances, albeit different.

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  • $\begingroup$ Thank you so much for your detailed reply :) However I've had cases where the relationship between a predictor and a dependent data was significant in a multiple regression but not significant in simple linear regression.. How would I interpret this? Would I try to find which other predictor is affecting the relationship and control for it? $\endgroup$ – Nieve K Feb 1 '16 at 6:23

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