6. Usefulness of the Model. Correlation and Determination - Yousef's Notes
6. Usefulness of the Model. Correlation and Determination

6. Usefulness of the Model. Correlation and Determination

Sometimes we can fit a model that is adequate because we have enough evidence to indicate that the true slope is not zero, i.e., the independent variable (x) and the dependent variable (y) have a linear (straight) relationship, but, even if this relationship exists, it may be very weak. When the relationship is very weak we may want to use another independent variable (if we have that information) to try to explain the behavior of the dependent variable.

Therefore, the model we propose will be more useful the stronger the relationship between these two variables.

In this section we will answer these two questions to determine how useful our model is:

  • How strong is the relationship between y and x? Correlation coefficient (r)
  • How much does the information provided by x contribute to describing or predicting y? Coefficient of determination ($r^2$)

#Correlation coefficient (r)

Answers “According to our sample, how strong is the linear relationship between two variables”?

  • The correlation coefficient measures the degree of association between the independent variable (x) and the dependent variable (y).
  • The possible values of this coefficient range from -1 to 1, depending on the strength and direction of the relationship between the two variables.
  • CAUTION: this coefficient does not indicate a cause-effect relationship.
  • If the correlation between x and y is weak, we may want to find another independent variable (x) if we want to get good results in the prediction of the dependent variable (y).

It seems that the information given by the correlation between two variables is the same as that obtained by making inferences about the true slope 𝜷1 . It is true that both metrics give us information about whether the relationship exists and the direction of the relationship, but:

  • The correlation coefficient (r) is a value we get from the sample, so we need to make inferences about the true slope 𝜷1 to describe the REAL relationship that exists between the two variables.
  • The value of the true slope 𝜷1 does not indicate the strength of the relationship between the two variables, that’s why we use r.

Strong when $r \geq 0.7$

#Coefficient of determination ($r^2$)

Another way to measure the usefulness of a linear model is to measure the contribution of x in predicting y. To accomplish this, we calculate how much the errors of prediction of y were reduced by using the information provided by x (as compared to the errors we would obtain if we used only the mean of y to predict the value of new observations)

$r^2$ represents the proportion of the total sample variability around ȳ that is explained by the linear relationship between x and y.

Note that $r^2$ is always between 0 and 1 because r is between -1 and +1. Thus, an $r^2$ of .60 indicates that the variation of the y values about their predicted values has been reduced 60% by the use of the least squares equation ŷ, instead of ȳ, to predict y.

#Calculating and Interpreting Usefulness of the Model

$r^2$: 81.67% of the sample variation in reaction time (y) can be ’explained’ by using the percent of drug in the blood (x) in a straight-line model.

r: According to our sample data, there is a correlation of 0.904 (very high) between the percentage of drug in blood (x) and the reaction time (y). This indicates that there is a strong positive relationship between the two variables in our sample (DOES NOT IMPLY CAUSATION).

Conclusion: the percentage of drug in blood (x) seems to be a good predictor of reaction time (y).