# High correlation but no linear relationship between x

### Introduction to Correlation and Regression Analysis

A coeffcient close to 0 means no linear relation between the two variables. is +/- and above, high degree of correlation or the association between the. Correlation between two variables indicates that a relationship exists between those variables. The extreme values of -1 and 1 indicate a perfectly linear relationship where a change in A coefficient of zero represents no linear relationship. For negative relationships, high values of one variable are associated with low. Measuring Linear Association: Correlation. ▫ Facts About Calculate and interpret correlation. ➢ Describe The most useful graph for displaying the relationship between two Individuals with higher GPAs are located further to the right and . 3) r has no units and does not change when the units of measure of x, y, or both.

In practice, meaningful correlations i.

### Correlation and dependence - Wikipedia

There are also statistical tests to determine whether an observed correlation is statistically significant or not i. Procedures to test whether an observed sample correlation is suggestive of a statistically significant correlation are described in detail in Kleinbaum, Kupper and Muller.

We introduce the technique here and expand on its uses in subsequent modules. Simple Linear Regression Simple linear regression is a technique that is appropriate to understand the association between one independent or predictor variable and one continuous dependent or outcome variable. In regression analysis, the dependent variable is denoted Y and the independent variable is denoted X. When there is a single continuous dependent variable and a single independent variable, the analysis is called a simple linear regression analysis.

This analysis assumes that there is a linear association between the two variables. If a different relationship is hypothesized, such as a curvilinear or exponential relationship, alternative regression analyses are performed. The figure below is a scatter diagram illustrating the relationship between BMI and total cholesterol. Each point represents the observed x, y pair, in this case, BMI and the corresponding total cholesterol measured in each participant.

Note that the independent variable BMI is on the horizontal axis and the dependent variable Total Serum Cholesterol on the vertical axis. BMI and Total Cholesterol The graph shows that there is a positive or direct association between BMI and total cholesterol; participants with lower BMI are more likely to have lower total cholesterol levels and participants with higher BMI are more likely to have higher total cholesterol levels.

### Scatter Plots and Linear Correlation ( Read ) | Statistics | CK Foundation

For either of these relationships we could use simple linear regression analysis to estimate the equation of the line that best describes the association between the independent variable and the dependent variable.

Example - Correlation of Gestational Age and Birth Weight A small study is conducted involving 17 infants to investigate the association between gestational age at birth, measured in weeks, and birth weight, measured in grams. We wish to estimate the association between gestational age and infant birth weight. In this example, birth weight is the dependent variable and gestational age is the independent variable.

The data are displayed in a scatter diagram in the figure below. Each point represents an x,y pair in this case the gestational age, measured in weeks, and the birth weight, measured in grams.

Note that the independent variable is on the horizontal axis or X-axisand the dependent variable is on the vertical axis or Y-axis. The scatter plot shows a positive or direct association between gestational age and birth weight.

Infants with shorter gestational ages are more likely to be born with lower weights and infants with longer gestational ages are more likely to be born with higher weights. The formula for the sample correlation coefficient is where Cov x,y is the covariance of x and y defined as are the sample variances of x and y, defined as The variances of x and y measure the variability of the x scores and y scores around their respective sample meansconsidered separately.

The covariance measures the variability of the x,y pairs around the mean of x and mean of y, considered simultaneously.

## Statistics review 7: Correlation and regression

To compute the sample correlation coefficient, we need to compute the variance of gestational age, the variance of birth weight and also the covariance of gestational age and birth weight. We first summarize the gestational age data. The mean gestational age is: To compute the variance of gestational age, we need to sum the squared deviations or differences between each observed gestational age and the mean gestational age.

The computations are summarized below. The variance of gestational age is: Next, we summarize the birth weight data. The mean birth weight is: The variance of birth weight is computed just as we did for gestational age as shown in the table below. The variance of birth weight is: Next we compute the covariance, To compute the covariance of gestational age and birth weight, we need to multiply the deviation from the mean gestational age by the deviation from the mean birth weight for each participant i.

Notice that we simply copy the deviations from the mean gestational age and birth weight from the two tables above into the table below and multiply. The covariance of gestational age and birth weight is: We now compute the sample correlation coefficient: Not surprisingly, the sample correlation coefficient indicates a strong positive correlation.