Dependent Variable A variable whose values are thought to be a function of, in understanding the bivariate relationship between two quantitative variables. DIAGRAMs PERSONAL COMPUTERS Can you think of any product that has The relationship between processor speed and price is somewhat curvilinear and. Variable 1 2 3 4 5 6 7 8 9 WrdlVlean 1 SntComp 1 OddWrds 1 explanatory variables (the factors) are unobserved variables whose values product— moment correlation measures the linear relationship between two variables. curvilinear relationship between two variables, but a researcher decides that. This article examines the relationships between interaction (product) terms and curvilinear (quadratic) relationship between two variables is indeed associ- ated with their product. .. educational expectations were measured) and whose.
Not surprisingly, the sample correlation coefficient indicates a strong positive correlation. In practice, meaningful correlations i.
Correlation and Linear Regression
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.Multiple Linear Regression: Curvilinear (quadratic & cubic) and Interaction
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. However, the correlation coefficient is not.
For samples, the correlation coefficient is represented by r while the correlation coefficient for populations is denoted by the Greek letter rho which can look like a p.
Be aware that the Spearman rho correlation coefficient also uses the Greek letter rho, but generally applies to samples and the data are rankings ordinal data. The closer r is to -1, the stronger the negative correlation is. Temperature in Celsius and Fahrenheit are perfectly correlated. Formal hypothesis testing can be applied to r to determine how significant a result is. That is the subject of Hinkle chapter 17 and this lesson The Student t distribution with n-2 degrees of freedom is used.
Remember, correlation does not imply causation. A value of zero for r does not mean that there is no correlation, there could be a nonlinear correlation.
Confounding variables might also be involved. Suppose you discover that miners have a higher than average rate of lung cancer.
You might be tempted to immediate conclude that their occupation is the cause, whereas perhaps the region has an abundance of radioactive radon gas leaking from the subterranian regions and all people in that area are affected. Or, perhaps, they are heavy smokers It is the fraction of the variation in the values of y that is explained by least-squares regression of y on x.
This will be discussed further in lesson 6 after least squares is introduced. Correlation coefficients whose magnitude are between 0.