SMHS SLR

Scientific Methods for Health Sciences - Correlation and Simple Linear Regression (SLR)

Overview

Many scientific applications involve the analysis of relationships between two or more variables involved in studying a process of interest. In this section, we are going to study on the correlations between 2 variables and start with simple linear regressions. Consider the simplest of all situations where Bivariate data (X and Y) are measured for a process and we are interested in determining the association with an appropriate model for the given observations. The first part of this lecture will discuss about correlation and then we are going to talk about SLR to address correlations.

Motivation

The analysis of relationships, if any, between two or more variables involved in the process of interest is widely needed in various studies. We begin with the simplest of all situations where bivariate data (X and Y) are measured for a process and we are interested in determining the association, relation or an appropriate model for these observations (e.g., fitting a straight line to the pairs of (X,Y) data). For example, we measured students of their math scores in the final exam and we want to find out if there is any association between the final score and their participation rate in the math class. Or we are interested to find out if there is any association between weight and lung capacity. Simple linear regression would certainly be a simple way to start and it can address the association very well in simple cases.

Theory

• Correlation: correlation efficient (-1≤ρ≤1) is a measure of linear association or clustering around a line of multivariate data. The main relationship between two variables (X,Y) can be summarized by (μ_X,σ_X ),(μ_Y,σ_Y) and the correlation coefficient denoted by ρ=ρ(X,Y).
  • The correlation is defined only if both of the standard deviations are finite and are nonzero and it is bounded by -1≤ρ≤1.
  • If ρ=1, perfect positive correlation (straight line relationship between the two variables); if ρ=0, no correlation (random cloud scatter), i.e., no linear relation between X and Y; if ρ=-1, a perfect negative correlation between the variables.
  • ρ$(X,Y)=\frac{cov(X,Y)}{\sigma_{X}\sigma_{Y}}$=$\frac{E((X-μ_{X})(Y-μ_{Y}))}{\sigma_{X}\sigma_{Y}}$=$\frac{E(XY)-E(X)E(Y)} {\sqrt{E(X^{2})-E^{2}(X)}\sqrt{E(Y^{2})-E^{2}(Y)}},$ where E is the expectation operator, and cov is the covariance. $μ_{X}=E(X),\sigma_{X}^{2}=E(X^{2})-E^{2}(X),$ and similarly for the second variable, Y, and $cov(X,Y)=E(XY)-E(X)*E(Y)$.
  • Sample correlation: replace the unknown expectations and standard deviations by sample mean and sample standard deviation: suppose ${X_{1},X_{2},…,X_{n}}$ and ${Y_{1},Y_{2},…,Y_{n}}$ are bivariate observations of the same process and (μ_X,σ_X ),(μ_Y,σ_Y) are the mean and standard deviations for the X and Y measurements respectively. $ρ(x,y)=\frac{\sum x_{i} y_{i}-n\bar{x}\bar{y}}{(n-1)s_{x} s_{y}}$=$\frac{n \sum x_{i} y_{i}-\sum x_{i}\sum y_{i}} {{\sqrt{n\sum x_{i}^{2} -(\sum x_{i})^{2}}} {\sqrt{ n\sum y_{i}^{2}-y_{i})^{2}}}}$
  • Example: Human weight and height (suppose we took only 6 of the over 25000 observations of human weight and height included in SOCR dataset .

Subject Index	Height $(x_{i})$ in cm	Weight $(y_{i})$ in kg	$x_{i}-\bar x$	$y_{i}-\bar y$	$(x_{i}-\bar x)^{2}$	$(y_{i}-\bar y)^{2}$	$(x_{i}-\bar x)(y_{i}-\bar y)$
1	167	60	6	4.6	36	21.82	28.02
2	170	64	9	8.67	81	75.17	78.03
3	160	57	-1	1.67	1	2.79	-1.67
4	152	46	-9	-9.33	81	87.05	83.97
5	157	55	-4	-0.33	16	0.11	1.32
6	160	50	-1	-5.33	1	28.41	5.33
Total	966	332	0	0	216	215.33	195

$\bar x\frac {966}{6}=161, \bar y=\frac {322}{6}= 55. s_{x}=\sqrt{\frac{216.5}{5}}=6.57, s_{y}=\sqrt{\frac {215.3}{5}}=6.56.$

$p(x,y)=\frac{1}{n-1}$$\sum\frac{x_{i}-\bar x}{s_{x}}\frac{y_{1}-\bar y}{s_{y}}=0.904$

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• SOCR Home page: http://www.socr.umich.edu

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