Question

This week's discussion is about correlation and regression concepts. Use the internet to find a website that shows an example or application of correlation or regression in an area of interest in your personal or professional life. Discuss how they used correlation or regression. Summarize your findings and share them. Be sure to define the independent and dependent variables. Also, discuss the impact/relevance of the independent variable.

Be sure to support your statements with logic and argument, citing any sources referenced.

Answer 1

Correlation is described as the analysis which lets us know the association or the absence of the relationship between two variables ‘x’ and ‘y’. On the other end, 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. For example, in students taking a Maths and English test, we could use correlation to determine whether students who are good at Maths tend to be good at English as well, and regression to determine whether the marks in English can be predicted for given marks in Maths. We can use the correlation coefficient, such as the Pearson Product Moment Correlation Coefficient to test if there is a linear relationship between the variables. To quantify the strength of the relationship, we can calculate the correlation coefficient (r). Its numerical value ranges from +1.0 to -1.0. r > 0 indicates positive linear relationship, r < 0 indicates negative linear relationship while r = 0 indicates no linear relationship. In regression analysis the problem of interest is the nature of the relationship itself between the dependent variable (response) and the (explanatory) independent variable. The independent variable is "independent" because its variation does not depend on the variation of another variable in the experiment/research project. In an experiment, the independent variable is manipulated and the effects observed. These observed effects are called dependent variables. The regression analysis consists of choosing and fitting an appropriate model, done by the method of least squares, with a view to exploiting the relationship between the variables to help estimate the expected response for a given value of the independent variable. For example, if we are interested in the effect of age on height, then by fitting a regression line, we can predict the height for a given age.

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. Thus y=birth weight and x=gestational age. The data are displayed in a scatter diagram in the figure below.

The independent variable is on the horizontal axis (or X-axis), and 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 means (

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

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: 49334/17= 2902 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.e.,

The computations are summarized below. 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.

As we noted, sample correlation coefficients range from -1 to +1. In practice, meaningful correlations (i.e., correlations that are clinically or practically important) can be as small as 0.4 (or -0.4) for positive (or negative) associations. There are also statistical tests to determine whether an observed correlation is statistically significant or not (i.e., statistically significantly different from zero). Procedures to test whether an observed sample correlation is suggestive of a statistically significant correlation are described in detail in Kleinbaum, Kupper and Muller.¹

EXAMPLE OF REGRESSION ANALYSIS

In the table below, the x_i column shows scores on the aptitude test. Similarly, the y_i column shows statistics grades. The last two columns show deviations scores - the difference between the student's score and the average score on each test. The last two rows show sums and mean scores that we will use to conduct the regression analysis And for each student, we also need to compute the squares of the deviation scores (the last two columns in the table below).

Student	xi	yi	(xi-x)2	(yi-y)2
1	95	85	289	64
2	85	95	49	324
3	80	70	4	49
4	70	65	64	144
5	60	70	324	49
Sum	390	385	730	630
Mean	78	77

And finally, for each student, we need to compute the product of the deviation scores.

Student	xi	yi	(xi-x)(yi-y)
1	95	85	136
2	85	95	126
3	80	70	-14
4	70	65	96
5	60	70	126
Sum	390	385	470
Mean	78	77

The regression equation is a linear equation of the form: ŷ = b₀ + b₁x . To conduct a regression analysis, we need to solve for b₀ and b₁. Computations are shown below. Notice that all of our inputs for the regression analysis come from the above three tables. First, we solve for the regression coefficient (b₁):

b₁ = Σ [ (x_i - x)(y_i - y) ] / Σ [ (x_i - x)²] ; b₁ = 470/730=0.644

Once we know the value of the regression coefficient (b₁), we can solve for the regression slope (b₀):

b₀ = y - b₁ * x

b₀ = 77 - (0.644)(78) = 26.768

Therefore, the regression equation is: ŷ = 26.768 + 0.644x .

In our example, the independent variable is the student's score on the aptitude test. The dependent variable is the student's statistics grade. If a student made an 80 on the aptitude test, the estimated statistics grade (ŷ) would be:

ŷ = b₀ + b₁x

ŷ = 26.768 + 0.644x = 26.768 + 0.644 * 80

ŷ = 26.768 + 51.52 = 78.288