# How is a relationship between two variables measured in statistics

### Statistical Language - Correlation and Causation

Relationships Between Variables, Part 3: Measures of Relationships. In this section, we discuss measures of relationships between two variables X and Y. It seems that a measure of a relationship should depend on what. Measuring Linear Association: Correlation The basic practice of statistics (6th relationship between two quantitative variables, it is always helpful to create. Answer to How is a linear relationship between two variables measured in statistics? Choose all The true statements. A. There are.

If there is a significant association between the two sets of ranks, health officials may feel more confident in their strategy than if a significant association is not evident. Chi-square test The chi-square test for association contingency is a standard measure for association between two categorical variables.

A simple and generic example follows. If scientists were studying the relationship between gender and political partythen they could count people from a random sample belonging to the various combinations: The scientists could then perform a chi-square test to determine whether there was a significant disproportionate membership among those groups, indicating an association between gender and political party.

Relative risk and odds ratio Specifically in epidemiology, several other measures of association between categorical variables are used, including relative risk and odds ratio. Relative risk is appropriately applied to categorical data derived from an epidemiologic cohort study. It measures the strength of an association by considering the incidence of an event in an identifiable group numerator and comparing that with the incidence in a baseline group denominator.

A relative risk of 1 indicates no association, whereas a relative risk other than 1 indicates an association. As an example, suppose that 10 out of 1, people exposed to a factor X developed liver cancerwhile only 2 out of 1, people who were never exposed to X developed liver cancer.

Thus, the strength of the association is 5, or, interpreted another way, people exposed to X are five times more likely to develop liver cancer than people not exposed to X. If the relative risk was less than 1 perhaps 0. The categorical variables are exposure to X yes or no and the outcome of liver cancer yes or no. This calculation of the relative risk, however, does not test for statistical significance. If the confidence interval does not include 1, the relationship is considered significant.

Similarly, an odds ratio is an appropriate measure of strength of association for categorical data derived from a case-control study. The odds ratio is often interpreted the same way that relative risk is interpreted when measuring the strength of the association, although this is somewhat controversial when the risk factor being studied is common. Additional methods There are a number of other measures of association for a variety of circumstances. Other combinations of data types or transformed data types may require the use of more specialized methods to measure the association in strength and significance.

Other types of association describe the way data are related but are usually not investigated for their own interest. Serial correlation also known as autocorrelationfor instance, describes how in a series of events occurring over a period of time, events that occur closely spaced in time tend to be more similar than those more widely spaced.

The Durbin-Watson test is a procedure to test the significance of such correlations. If the correlations are evident, then it may be concluded that the data violate the assumptions of independence, rendering many modeling procedures invalid.

A classical example of this problem occurs when data are collected over time for one particular characteristic. It is easiest to start with no relationship. What do we mean by no relationship?

## Australian Bureau of Statistics

Suppose we had a lot of data on X,Y and obtained a scatterplot of Y versus X. If the plot was a random scatter then we would conclude that the variables X and Y are not related. What if they are related? In the first, we would probably conclude that X and Y are not related. Plot 2, we would characterize as probably a linear relationship, certainly exhibiting random error. Plot 3 is similar to Plot 2, although the pattern is not quite as tight. Plot 4 shows some negative drift. Plots 5 and 6 show the strongest relationships tightest patterns among the plots.

Plot 5 shows a very strong circular relationship while Plot 6 a very strong quadratic pattern. It seems that a measure of a relationship should depend on what type of relationship it is. In this section, we will only be concerned for the most part about linear relationships and we will consider measures of such a relationship.

It should not be surprising that this measure will indicate no linear relationship for the two strongest relationships in the plots. Scatter plots Consider Plot 2 again. We want to measure the linear relationship exhibited in this plot. Two simple lines will help a lot. On the x-axis locate the sample mean of the X's and draw a vertical line through this point.

On the y-axis locate the sample mean of the Y's and draw a horizontal line through this point. Plot 2 with sample means The lines intersect atlocate it.

This is our new center. The coordinates of X,Y relative to the new center are. Then it's easy to come up with many measures of linear relationships.

### Relationships Between Two Variables | STAT

A simple one is to count the number of points with the same sign those in quadrants I and III and subtract the number of points with different signs those in quadrants II and IV.

High values of this measure indicate a positive linear relationship while low values indicate a negative linear relationship. Instead of counting like and unlike signs, we consider a measure which takes the product of these new coordinates. Thus we have n products, one for each point in the plot. Consider as a measure their average: Positive values of this measure indicate a positive linear relationship while negative values indicate a negative linear relationship.

## How is a linear relationship between 2 variables measured in statistics?

Is this measure robust? No, you are catching on. For a given data set, we can always make this measure larger or smaller by changing the units.

Suppose we have a positive linear relationship and X is measured in feet. If we change the X's to inches then sXY increases by the factor If we change the X's to mm's then sXY increases by the factor