regression and correlation

Correlation and Regression


The test you choose depends on level of measurement:

Independent Dependent Test

Dichotomous Interval-Ratio Independent Samples t-testDichotomous

Nominal Interval-Ratio ANOVADichotomous Dichotomous

Nominal Nominal Cross TabsDichotomous Dichotomous

Interval-Ratio Interval-Ratio Bivariate Regression/CorrelationDichotomous


Bivariate regression is a technique that fits a straight line as close as possible between all the coordinates of two continuous variables plotted on a two-dimensional graph--to summarize the relationship between the variables

Correlation is a statistic that assesses the strength and direction of association of two continuous variables . . . It is created through a technique called “regression”

Bivariate Regression

For example:A criminologist may be interested in the relationship between Income and Number of Children in a family or self-esteem and criminal behavior.

Independent Variables

Family Income

Self-esteem

Dependent Variables

Number of Children

Criminal Behavior


For example: Research Hypotheses:

As family income increases, the number of children in families declines (negative relationship).

As self-esteem increases, reports of criminal behavior increase (positive relationship).


Family Income

Self-esteem

Dependent Variables

Number of Children

Criminal Behavior


For example: Null Hypotheses:

There is no relationship between family income and the number of children in families. The relationship statistic b = 0.

There is no relationship between self-esteem and criminal behavior. The relationship statistic b = 0.


Family Income

Self-esteem

Dependent Variables

Number of Children

Criminal Behavior

Bivariate Regression Let’s look at the relationship between self-esteem

and criminal behavior. Regression starts with plots of coordinates of

variables in a hypothesis (although you will hardly ever “plot” your data in reality).

The data:

Each respondent has filled out a self-esteem assessment and reported number of crimes committed.

Bivariate RegressionY, crimes

X, self-esteem

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What do you think the relationship is?

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Is it positive?

Negative?

No change?

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Regression is a procedure that fits a line to the data. The slope of that line acts as a model for the relationship between the plotted variables.

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The slope of a line is the change in the corresponding Y value for each unit increase in X (rise over run).

Slope = 0, No relationship!

Slope = 0.2, Positive Relationship!

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Slope = -0.2, Negative Relationship!

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The mathematical equation for a line:

Y = mx + b

Where: Y = the line’s position on the vertical axis at any point

X = the line’s position on the horizontal axis at any point

m = the slope of the line

b = the intercept with the Y axis,

where X equals zero


The statistics equation for a line:

Y = a + bx

Where: Y = the line’s position on the vertical axis at any point (value of dependent variable)

X = the line’s position on the horizontal axis at any point (value of the independent variable)

b = the slope of the line (called the coefficient)

a = the intercept with the Y axis,

where X equals zero

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The next question:

How do we draw the line???

Our goal for the line:

Fit the line as close as possible to all the data points for all values of X.


X, self-esteem

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How do we minimize the distance between a line and all the data points?

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• How do we minimize the distance between a line and all the data points?

• You already know of a statistic that minimizes the distance between itself and all data values for a variable--the mean!

• The mean minimizes the sum of squared deviations--it is where deviations sum to zero and where the squared deviations are at their lowest value. (Y - Y-bar)2

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• The mean minimizes the sum of squared deviations--it is where deviations sum to zero and where the squared deviations are at their lowest value.

• Take this principle and “fit the line” to the place where squared deviations (on Y) from the line are at their lowest value (across all X’s).

(Y - Y)2 Y = line ^ ^


There are several lines that you could draw where the deviations would sum to zero...

Minimizing the sum of squared errors gives you the unique, best fitting line for all the data points. It is the line that is closest to all points.

Y or Y-hat = Y value for line at any X Y = case value on variable Y Y - Y = residual (Y – Y) = 0; therefore, we use (Y - Y)2 …and

minimize that!

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Illustration of Y – Y

= Yi, actual Y value corresponding w/ actual X

= Yi, line level on Y corresponding w/ actual X

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Y = 10, Y = 5

Y = 0, Y = 4

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Illustration of (Y – Y)2



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(Yi – Y)2 = deviation2 Y = 10, Y = 5 . . . 25

Y = 0, Y = 4 . . . 16

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Illustration of (Y – Y)2



The goal: Find the line that minimizes sum of deviations squared.

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The best line will have the lowest value of sum of deviations squared (adding squared deviations for each case in the sample.


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Y = a + bX∧

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e


We use (Y - Y)2 …and minimize that! There is a simple, elegant formula for “discovering”

the line that minimizes the sum of squared errors

((X - X)(Y - Y))

b = (X - X)2 a = Y - bX Y = a + bX This is the method of least squares, it gives our

least squares estimate and indicates why we call this technique “ordinary least squares” or OLS regression

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Bivariate RegressionY

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Considering that a regression line minimizes (Y - Y)2, where would the regression line cross for an interval-ratio variable regressed on a dichotomous independent variable?

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For example:

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1=Women: Mean = 4

Bivariate RegressionY

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The difference of means will be the slope. This is the same number that is tested for significance in an independent samples t-test.

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0=Men: Mean = 6

1=Women: Mean = 4

Slope = -2 ; Y = 6 – 2X

Correlation This lecture has

covered how to model the relationship between two variables with regression.

Another concept is strength of association.

Correlation provides that.

CorrelationY, crimes

X, self-esteem

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So our equation is:

Y = 6 - .2X

The slope tells us direction of association… How strong is that?

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Correlation1

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Example of Low Negative Correlation

When there is a lot of difference on the dependent variable across subjects at particular values of X, there is NOT as much association (weaker).

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Example of High Negative Correlation

When there is little difference on the dependent variable across subjects at particular values of X, there is MORE association (Stronger).

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Correlation

To find the strength of the relationship between two variables, we need correlation.

The correlation is the standardized slope… it refers to the standard deviation change in Y when you go up a standard deviation in X.

Correlation

The correlation is the standardized slope… it refers to the standard deviation change in Y when you go up a standard deviation in X.

(X - X)2

Recall that s.d. of x, Sx = n - 1

(Y - Y)2

and the s.d. of y, Sy = n - 1

Sx Pearson correlation, r = Sy b

Correlation

The Pearson Correlation, r: tells the direction and strength of the relationship between

continuous variablesranges from -1 to +1 is + when the relationship is positive and - when the

relationship is negative the higher the absolute value of r, the stronger the

associationa standard deviation change in x corresponds with r

standard deviation change in Y

Correlation

The Pearson Correlation, r:The pearson correlation is a statistic that is an

inferential statistic too.

r - (null = 0) tn-2 = (1-r2) (n-2)

When it is significant, there is a relationship in the population that is not equal to zero!

Error Analysis

Y = a + bX This equation gives the conditional mean of Y at any given value of X.

So… In reality, our line gives us the expected mean of Y given each value of X The line’s equation tells you how the mean on your dependent variable changes

as your independent variable goes up.

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Error Analysis

As you know, every mean has a distribution around it--so there is a standard deviation. This is true for conditional means as well. So, you also have a conditional standard deviation.

“Conditional Standard Deviation” or “Root Mean Square Error” equals “approximate average deviation from the line.”

SSE ( Y - Y)2

= n - 2 = n - 2

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Y

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Error Analysis The Assumption of Homoskedasticity:

The variation around the line is the same no matter the X. The conditional standard deviation is for any given value of X.

If there is a relationship between X and Y, the conditional standard deviation is going to be less than the standard deviation of Y--if this is so, you have improved prediction of the mean value of Y by looking at each level of X.

If there were no relationship, the conditional standard deviation would be the same as the original, and the regression line would be flat at the mean of Y.

Y

X

Y Conditional standard deviation

Original standard deviation

Error Analysis

So guess what?

We have a way to determine how much our understanding of Y is improved when taking X into account—it is based on the fact that conditional standard deviations should be smaller than Y’s original standard deviation.

Error Analysis

Proportional Reduction in Error Let’s call the variation around the mean in Y “Error 1.” Let’s call the variation around the line when X is considered

“Error 2.”

But rather than going all the way to standard deviation to determine error, let’s just stop at the basic measure, Sum of Squared Deviations.

Error 1 (E1) = (Y – Y)2 also called “Sum of Squares”

Error 2 (E2) = (Y – Y)2 also called “Sum of Squared Errors”

Y

X

Y Error 2Error 1

R-Squared

Proportional Reduction in Error To determine how much taking X into consideration reduces the

variation in Y (at each level of X) we can use a simple formula:E1 – E2 Which tells us the

proportion or E1 percentage of original

error that is Explained by X.

Error 1 (E1) = (Y – Y)2

Error 2 (E2) = (Y – Y)2

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Error 2

Error 1

R-squared

r2 = E1 - E2 E1

= TSS - SSE TSS

= (Y – Y)2 - (Y – Y)2

(Y – Y)2

r2 is called the “coefficient of determination”…

It is also the square of the Pearson correlation

Y

X

Y Error 2

Error 1

R-Squared R2:

Is the improvement obtained by using X (and drawing a line through the conditional means) in getting as near as possible to everybody’s value for Y over just using the mean for Y alone.

Falls between 0 and 1 Of 1 means an exact fit (and there is no variation of scores

around the regression line) Of 0 means no relationship (and as much scatter as in the

original Y variable and a flat regression line through the mean of Y)

Would be the same for X regressed on Y as for Y regressed on X

Can be interpreted as the percentage of variability in Y that is explained by X.

Some people get hung up on maximizing R2, but this is too bad because any effect is still a finding—a small R2 only indicates that you haven’t told the whole (or much of the) story with your variable.

Error Analysis, SPSSSome SPSS output (Anti- Gay Marriage regressed on Age):

r2

(Y – Y)2 - (Y – Y)2 (Y – Y)2

196.886 ÷ 2853.286 = .069

Line to the Mean

Data points to the lineData points to the mean

Original SS for Anti- Gay Marriage

Error AnalysisSome SPSS output (Anti- Gay Marriage regressed on Age):

r2

(Y – Y)2 - (Y – Y)2 (Y – Y)2

196.886 ÷ 2853.286 = .069

Line to the Mean

Data points to the lineData points to the mean

0 18 45 89 Age

Strong Oppose 5

Oppose 4

Neutral 3

Support 2

Strong Support 1

Anti- Gay MarriageM = 2.98

Colored lines are examples of:Distance from each person’s data point to the line or model—new, still unexplained error.Distance from line or model to Mean for each person—reduction in error.Distance from each person’s data point to the Mean—original variable’s error.

ANOVA Table

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YMean

Q: Why do I see an ANOVA Table?

A: We “bust up” variance to get R2. Each case has a value for distance from the line (Y-barcond. Mean) to Y-barbig, and a value for distance from its Y value and the line (Y-barcond. Mean).

Squared distance from the line to the mean (Regression SS) is equivalent to BSS, df = 1. In ANOVA, all in a group share Y-bargroup

The squared distance from the line to the data values on Y (Residual SS) is equivalent to WSS, df = n-2. In ANOVA, all in a group share Y-bargroup

The ratio, Regression to Residual SS, forms an F distribution in repeated sampling. If F is significant, X explains some variation in Y.

BSS

WSS

TSS

Line Intersects Group Means

Dichotomous VariablesY

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Using a dichotomous independent variable, the ANOVA table in bivariate regression will have the same numbers and ANOVA results as a one-way ANOVA table would (and compare this with an independent samples t-test).

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0=Men: Mean = 6

1=Women: Mean = 4

Slope = -2 ; Y = 6 – 2X

Mean = 5 BSS

WSS

TSS

Regression, Inferential Statistics

Descriptive: The equation for your line

is a descriptive statistic. It tells you the real, best-fitted line that minimizes squared errors.

Inferential: But what about the

population? What can we say about the relationship between your variables in the population???

The inferential statistics are estimates based on the best-fitted line.

Recall that statistics are divided between descriptive and inferential statistics.

Regression, Inferential Statistics

The significance of F, you already understand.

The ratio of Regression (line to the mean of Y) to Residual (line to data point) Sums of Squares forms an F ratio in repeated sampling.

Null: r2 = 0 in the population. If F exceeds critical F, then your variables have a relationship in the population (X explains some of the variation in Y).

Most extreme 5% of F’s

F = Regression SS / Residual SS

Regression, Inferential Statistics What about the Slope or

“Coefficient”? From sample to sample, different

slopes would be obtained. The slope has a sampling

distribution that is normally distributed.

So we can do a significance test. -3 -2 -1 0 1 2 3 z

Regression, Inferential StatisticsConducting a Test of Significance for the slope of the Regression Line

By slapping the sampling distribution for the slope over a guess of the population’s slope, Ho, one determines whether a sample could have been drawn from a population where the slope is equal Ho.

1. Two-tailed significance test for -level = .052. Critical t = +/- 1.963. To find if there is a significant slope in the population,

Ho: = 0Ha: 0 ( Y – Y )2

n Collect Data n - 2n Calculate t (z): t = b – o s.e. = s.e. ( X – X )2

n Make decision about the null hypothesisn Find P-value

Correlation and RegressionBack to the SPSS output:

The standard error and t appears on SPSS output …and the p-value too!

Correlation and RegressionBack to the SPSS output:

Y = 1.88 + .023X

So the GSS example, the slope is significant. There is evidence of a positive relationship in the population between Age and Anti- Gay Marriage sentiment. 6.9% of the variation in Marriage attitude is explained by age. The older Americans get, the more likely they are to oppose gay marriage.

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A one year increase in age elevates “anti” attitudes by .023 scale units. There is a weak positive correlation. A s.d, increase in age produces a .023 s.d. increase in “anti” scale units.

regression and correlation

Leadership & Management

bivariate regression

positive relationship

negative relationship

variables correlation

relationship statistic

continuous variables

plotted variables

selfesteem increases