# Why don't the results of testing $H_0 : \beta = 0$ and $H_0 : {\rm cor}(X,Y)=0$ agree?

I have 4 IVs in my model that directly effect the DV. The results of the correlation & regression analyses showed that:

IV1&DV:

Pearson Correlation Coefficient: insignificant

Regression Beta&t values: significant

IV2&DV:

Pearson Correlation Coefficient: significant

Regression Beta&t values: insignificant

IV3&DV:

Pearson Correlation Coefficient: significant

Regression Beta&t values: significant

IV4&DV:

Pearson Correlation Coefficient: significant

Regression Beta&t values: significant

No multicollinearity problem was detected and also all the regression assumptions have been perfectly met!

Why the results of the regression analysis turned out to be completely opposite of correlation analysis for IV1 and IV2?!! Why they're being contradictory?! Does it make any sense? Is it acceptable? TQ.

• Do you have a single model with the 4 covariates? If so which Pearson correlation are you talking about. There are different one's. You may be looking at conditional correlations. I think you are too vague. Please describe the problem in more details and tell us which correlations you are comparing to which betas? – Michael R. Chernick Jul 24 '12 at 11:15
• Yes, it's a single model with a DV and 4 IVS. I obtained the Pearson correlation coefficients between DV&IVs in spss from Analyze>Correlate>Bivariate. And by beta I mean the1 that is in regression "coefficients" table. – Cyrus Jul 24 '12 at 11:35
• @ Michael: U can find the results here. – Cyrus Jul 24 '12 at 12:02
• this thread may also be of interest. – Macro Jul 24 '12 at 17:51

The reason is that you're testing two different hypotheses:

• the Pearson correlation test is testing whether there is a non-zero correlation between the given predictor and the response variable, not taking into account the context supplied by the other predictors.

• The $t$-test for the regression coefficient is testing whether that predictor has a non-zero effect when the other predictors are in the model.

The two need not agree when some of the predictive power of a given predictor is subsumed another predictor (or predictors). The often happens when there is collinearity. For example, suppose that you have two predictors $X_1, X_2$ that are highly correlated with each other and are also highly correlated with the response, $Y$. Then it is quite likely that both will produce a significant result from the Pearson correlation test but, most likely, only one (or neither) of the two predictors will be significant when you enter them into the model simultaneously. Here is an example in R (unnecessary output lines were deleted):

x1 = rnorm(200)
x2 = .9*x1 + sqrt(1-.9^2)*rnorm(200)
y = 1 + 2*x1 + rnorm(200,sd=5)

# Pearson correlation test.
cor.test(x1,y)$p.value [1] 6.002424e-07 cor.test(x2,y)$p.value
[1] 3.473047e-07

# linear regression
summary(lm(y~x1+x2))
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)   1.3835     0.3445   4.016  8.4e-05 ***
x1            0.8621     0.8069   1.068    0.287
x2            1.1716     0.7893   1.484    0.139


What you may be thinking of is that when you're fitting a simple linear regression model, i.e. a regression with only one predictor, the Pearson correlation test will agree with the $t$-test of the regression coefficient:

summary( lm(y~x1) )
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)   1.3369     0.3441   3.886 0.000139 ***
x1            1.9249     0.3731   5.159    6e-07 ***


In that case, they actually are testing the same hypothesis - i.e. "is $X_1$ linearly related to $Y$?" - and it turns out that the hypothesis tests are actually exactly the same, so the $p$-values will be identical.

• Thanks Macro. Actually u r right, when i regressed the DV only on IV1 the Beta turned out insignificant. But at the presence of other 3 IVs the effect is significant! Seems interesting! – Cyrus Jul 24 '12 at 13:06
• I agree with with Macro's answer but I think the explanation might vary slightly if you were looking at partial correlations which takes account of the interaction between covariates. From the output it looks like the correlations are unconditional and taken from the correlation matrix for the 4 IVs and the DV. The IV are highly correlated with each other except for IV4 with IV1. – Michael R. Chernick Jul 24 '12 at 13:26