The descriptive statistics can be used to assist in calculating the correlation.
> lapply(CorrelationData, function(x) c(length(x), mean(x), sd(x)))
$Outcome1
[1] 4.00000 2.00000 2.44949
$Outcome2
[1] 4.00000 6.00000 2.44949
> cov(Outcome1,Outcome2)
[1] 3
The table of inferential statistics shows the key elements to be calculated.
> model <- lm(Outcome2 ~ Outcome1)
> summary(model)
Call:
lm(formula = Outcome2 ~ Outcome1)
Residuals:
1 2 3 4
-1.0 2.0 -2.5 1.5
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.0000 1.7854 2.801 0.107
Outcome1 0.5000 0.6124 0.816 0.500
Residual standard error: 2.598 on 2 degrees of freedom
Multiple R-squared: 0.25, Adjusted R-squared: -0.125
F-statistic: 0.6667 on 1 and 2 DF, p-value: 0.5
Descriptive Statistics: The descriptive statistics are calculated separately for each variable.
Sum of Cross Products: The Sum of Cross Products (SCP) is not easily determined solely from the summary statistics of the output, but rather from the data.
\[SCP = \sum ( X - M_X ) ( Y - M_Y ) = ( 0 - 2.000 ) ( 4 - 6.000 ) + ( 0 - 2.000 )( 7 - 6.000 ) + ( 3 - 2.000 )( 4 - 6.000 ) + (5 - 2.000)(9 - 6.000) = 9.000\]
Covariance: The Covariance is a function of the Sum of Cross Products and the sample size:
\[COV = \frac{SCP}{(N - 1)} = \frac{9.000}{(4 - 1)} = 3.000\]
Unstandardized Regression Coefficients: The Unstandardized Regression Coefficients involve Covariance, the Standard Deviations of the variables, and the Means of the variables:
\[B_1 = \frac{COV}{(SD)^2} = \frac{3.000}{(2.449)^2} = 0.500\]
\[B_0 = M_Y - (B_1)(M_X) = 6.000 - (0.500)(2.000) = 5.000\]
Standardized Regression Coefficients: The Standard Regression Coefficients involve the Regression Coefficient (for the predictor) and the Standard Deviations of the variables:
\[\beta_1 = (B_1)\frac{SD_X}{SD_Y} = (0.500)\frac{2.449}{2.449} = 0.500\]
Multiple Correlation: In bivariate regression, the Multiple Correlation is the same as the bivariate Correlation:
\[r = \frac{COV}{(SD_X) (SD_Y)} = \frac{3.000}{(2.449) (2.449)} = .500\]
Proportion of Variance Accounted For: The Proportion of Variance Accounted For is a function of the Correlation:
\[R^2 = .0500^2 = 0.250\]
Regression cofficients provide a measure of statistical relationship between two variables.
For the participants (N = 4), the scores on Outcome 1 (M = 2.00, SD = 2.45) moderately predicted Outcome 2 (M = 6.00, SD = 2.45), β = .500, R2= .250.
Note that regression coefficients can also have inferential information associated with them (and that this information should be summarized if it is available and of interest).
For the participants (N = 4), the scores on Outcome 1 (M = 2.00, SD = 2.45) did not significantly predict Outcome 2 (M = 6.00, SD = 2.45), β = .500, t = 0.816, p = .500.