Correlation versus regression

Correlation and Regression are the two analysis based on multivariate distribution. A multivariate distribution is described as a distribution of multiple variables.

• Correlation is described as the analysis which lets us know the association or the absence of the relationship between two variables ‘x’ and ‘y’.
• Regression analysis predicts the value of the dependent variable based on the known value of the independent variable, assuming that average mathematical relationship between two or more variables.

Correlation

A correlation measures three characteristics of the association between X and Y:

1. The direction of the relation. A positive correlation (+) emerges when two variables are moving in the same direction. If the value of X increases (for example the length of a person), the value of Y also increased (for example the weight of a person). A negative correlation occurs when two variables are moving in different directions. If X increases, Y decreases (or vice versa).

2. The form of the association. It can be for example linear.

3. The degree of the association. A perfect correlation has a value of -1 or 1. A correlation of 0 implies that there is no association between the two variables. A correlation of 0.8 is therefore stronger than a correlation of for example 0.5

Pearson correlation

The most well-known measure for correlation is the Pearson correlation. This correlation measures the degree and direction of a linear relation between two variables. The Pearson correlation is denoted with r and calculated as follows:

\[r = \frac{covariance\:of\:x\:and\:y}{variability\:of\:x\:and\:y\:seperately}\]

or:

\[r = \frac{\sum{(x-\bar{x})(y-\bar{y})}}{\sqrt{\sum{(x-\bar{x})^2} \sum{(y-\bar{y})^2}}}\]

which is the same as:

\[r = \frac{N \sum{xy}-(\sum{x})(\sum{y})}{\sqrt{[N\sum{x^2}-(\sum{x})^2] [N\sum{y^2}-(\sum{y})^2]}}\]

• N : number of pairs of scores
• x : x scores
• y : y scores
• x̄ : mean of x scores
• ȳ : mean of y scores

The Pearson correlation can also be calculated for z-scores with the following formula:

\[r = \frac{\sum{(z_x\cdot z_y)}}{N}\]

• z_y : z-score of x
• z_y : z-score of y
• N : number of pairs of scores

The proportion explained variance

With the Pearson correlation itself, you can not do so much, because it is not ratio scaled and thus not suitable for calculations. Therefore, you have to multiply it. The value r² is called the coefficient of determination. This value measures the proportion of variance within one variable, that can be explained by the association of this variable with another variable. A correlation of 0.80 (r = 0.80) implies for example that 0.64 (r²), that is 64% of the variance of scores on Y can be explained by variable X.

• An r² of 0.01 refers to a small correlation;
• An r² of 0.09 refers to a medium correlation;
• A large correlation is characterized by an r² of 0.25 or higher.

Spearman correlation

The Pearson correlation quantifies the linear relation between two variables. This correlation measure is used primarily when data are interval or ratio scaled - other correlation measures are developed for non-linear relations and other measurement scales.

The Spearman correlation measures the relation between two variables with an ordinal scale. The Spearman correlation can also be used when data are interval- or ratio scaled and there is no linear relation between X and Y.

The Spearman correlation looks for a consistent relation between X and Y, regardless of its form. The original values have to be ordered (from small to large). The Spearman correlation can be calculated as follows:

\[\rho = r_s = 1 - \frac{6\sum{d^2_i}}{n(n^2-1)}\]

• p : Spearman correlation
• r_s : Spearman correlation
• d_i : rg(X_i) - rg(Y_i): difference between the two ranks of each observation (for example, one can have the second best score on variable X, but the ninth on variable Y.)
• n : number of scores

The point-biserial correlation

A special variant of the Pearson correlation is called the point-biserial correlation. This correlation is used when one variable consists of number, but the other variable consists only of two categories. A variable with only two categories is called a dichotomous variable. An example is gender.

To calculate the point-biserial correlation, the dichotomous variable first has to be transferred to a variable with numerical values. One value (for example women) receives a zero and the other value (for example men) receives a one. Next, the formula for Pearson r is used.

The phi-coefficient

The phi-coefficient measures the relation between two variables that are both dichotomous. To do so, first the values 0 and 1 have to be given to both variables. Next, the Pearson r formula can be applied.

Strong and weak correlations

For large samples, even very small correlation may become statistically significant quickly. A significant correlation tells us nothing more than that the chance is very small that the correlation in the population equals zero. The presence of significance thus does not imply whether the relation between the variables is strong. The strength of a correlation is in accordance with the size of the correlation and not with the statistical significance of the correlation. The rule-of-thumb is that a correlation of .10 is weak, a correlation of .30 is moderate, and a correlation of .50 is strong.

Scatterplot

A useful way to examine the relation between two quantitative variables is a scatter plot. Each participants is displayed by a dot with coordinates, that refer to the values on the variables X and Y. Normally, the predictive variable is presented on the X-axis and the criterion variable is presented on the Y-axis. The criterion variable is predicted from the predictor variable. However, if it concerns a correlation coefficient, it is not always clear which variable is X and which is Y. In that case, it does not matter how the variables are labelled. In a scatterplot, a line is drawn through the cloud of dots as best as possible. That line is called the regression line of Y predicted by X (that is: Y on X) which gives the best approximation of Yi for a value Xi. If the regression line is straight, the relation between the variables is linear. If the regression line is curved, it is called a curvilinear relation.

The degree to which the dots lie around the regression line is related to the correlation (r) between X and Y. The closer the dots (the observed results) lie around the regression line (the predicted results), the higher the correlation. The correlation coefficient ranges from -1 to +1, in which a perfect correlation (all points are on the regression line) is referred with 1. The plus and minus sign indicate the direction and do not influence the relation between the variables.

Regression

The general formula for a simple regression is:

\[Y = a + bX + e\]

• Y : dependent variable
• X : independent variable
• a : the intercept (the value of y when x = 0)
• b : the slope of the regression line
• e : error, or the difference between the estimated and observed value of Y

For example, you have to pay 5 euros per hour next to the 30 euros entrance fee for a tennis club. You stay for 3 hours. In this case, the regression formula is:

• Y : dependent variable: how much you should pay
• X: independent variable: number of hours
• a : the intercept: entrance fee
• b : slope of the regression line: euros per hour
• e : error: additional tips that you would like to give

\[Y = a + bX + e = 30 + 5 \times 3 + tips = 45 + tips\]

Assumptions for regression

A few assumptions have to be met. First, there has to be homogeneity of variances. That means that the variance of Y is the same for each value of X in the population. In addition, the values of Y that are in accordance with the X-values have to be normally distributed.

When examining the sample correlation, we replace the regression model assumptions with the assumption that we draw a sample from a bivariate normal distribution. The conditional distributions in this distribution are the distributions of Y and X given a specific value of X or Y. When we look at all Y-values, independent of X, we call it the marginal distribution of Y. Finally, we assume that the relation between X and Y is linear.

Predicted values

To determine how well a line fits the data, we first have to calculate the distance between the line and each data point. For each X-value, the linear regression line determines the value for the Y variable. This value is called the predicted value (Ŷ). The distance between this predicted value and the actual Y-value is determined by the following steps:

1. Distance = Y - Ŷ. This distance measures the error between the line and the actual data.

2. Because some distances are negative, and others are positive, the next step is to square each distance, so that only positive values remain.

3. Finally, the total distance between the line and the data has to be calculated, which is called the . The squared values from step 2 are summed up: ∑(Y - Ŷ)². This is called the total squared error.

\[Total\:squared\:error = \sum{(Y - \hat{Y})^2}\]

• Y : actual value of Y
• Ŷ : predicted value of Y

Standardized regression coefficients

When the data is standardized, a difference of one unit in X refers to a difference of one standard deviation. If the slope is for example 0.75 (for standardized data), the Y will increase with 0.75 for each increase of one standard deviation of X. The slope of standardized data is called standardized regression coefficient or β.

For standardized data, it applies that sX = sY = s2X = 1, in which the slope and correlation coefficient are equal. A correlation of r = .80 implies that an increase of one standard deviation of X is associated with 8/10 standard deviation increase of Y. However, because it is a correlational association, we can not make claims about cause-and-effect.

Hypothesis tests for regression

The significance of b

If X and Y correlate, and there is a linear relation, the slope of the regression line will not be equal to zero and b will have a value different from zero. This is the case for one predictor variable, but when there are multiple predictor variables, the slope does not have to be significant for each of these variables.

b* is the parametric equivalent of b, namely the slope if we had X and Y measures on the whole population.

The standard error is:

\[s_b = \frac{Y - X}{X \sqrt{N - 1}}\]

• s_b : standard error
• Y : value of Y measure
• X : value of X measure
• N : number of measures