REVIEW Hypothesis testing starts with a null hypothesis

and a null distribution. We compare what we have to the null distribution,

if the result is too extreme to belong to the null distribution (p<.05), we reject the null hypothesis. Or this means the result supports our research/alternative hypothesis.

Test the assumption before you do the independent t-test. This affects which row of the statistical output to read for your decision-making.

Note: You DON’T have to have equal sample size for independent t-test.

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CHAPTER 9: CORRELATION

Day 6 Summer 2017.08.03

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DECISION TREE 8/2/2017

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RESEARCH ON RELATIONSHIP

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OVERVIEW

Scatter plots Correlation & covariance Various correlation coefficients Significance & null hypotheses Read SPSS output

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SCATTER PLOT: THE GRAPHIC REPRESENTATION A scatter plot is a graphical representation of the

relationship between two variables. Each subject or observation must have a pair of values,

one for X, one for Y.

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X : parent height Y : children’s height

We would like to see if

the children’s height is related to their parent’s height

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SCATTER PLOTS: POSITIVE RELATIONSHIP As X increase, Y also increases.

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r=.81

SCATTER PLOTS: NEGATIVE RELATIONSHIPS As X increase, Y decreases.

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r=-.79

SCATTER PLOTS: NO RELATIONSHIP

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r=.14

RELATIONSHIP DOES NOT HAVE TO BE LINEAR

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ANSCOMBE’S QUARTET

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SPSS DEMO Parent and child height data Select your X and Y variable:

sometimes it matters Incidence of cancer would be dependent on amount smoked,

not amount smoked dependent on cancer, thus it is more reasonable that amount smoked is selected as X, incidence of cancer is selected as Y.

sometimes it does not

Scatter plot Edit it Read the SPSS output

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COVARIANCE

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NXXXX

NXX

X∑∑ −−

=−

=))(()( 2

2σ

NYYXX

XY∑ −−

=))((

cov

Variance is defined as :

Covariance is defined as:

CORRELATION COEFFICIENTS

A correlation coefficient standardizes the covariance. We’ve seen the same thing before with z-scores Centered at 0.0, and divided by σ

Pearson product-moment correlation coefficient is defined as

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∑∑∑

∑∑

∑

−−

−−=

−−

−−

==2222 )()(

))((

)()(

))((cov

YYXX

YYXX

NYY

NXX

NYYXX

ssr

YX

XY

CORRELATION COEFFICIENTS

A correlation coefficient ranges between –1.0 and +1.0 Coefficient of 0 = no relationship between two

variables Negative values reflect a negative relationship Positive values reflect a positive relationship

The population coefficient is notated with ρ (rho)

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SEEING STATISTICS Drop the dots wherever you want and see how r

changes right away as you add more data.

http://www.uvm.edu/~dhowell/fundamentals8/SeeingStatisticsApplets/CorrelationPoints.html

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CORRELATION COEFFICIENTS

There are several correlation coefficients used by researchers

Common ones are: Pearson product-moment correlation (r) Spearman correlation (rs)

Different coefficients are appropriate for different situations.

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CORRELATION COEFFICIENTS

X

Y

Continuous Dichotomous Ranked

Continuous Pearson’s r Point-biserial

Dichotomous Point-biserial Phi (φ)

Ranked Spearman

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REMEMBER PEARSON’S R FOR TWO CONTINUOUS VARIABLES Defined as:

The closer to +1 or -1, the stronger the relationship The closer to 0, the weaker the relationship Some rules of thumb (very general)

Correlation between about 0.0 to 0.1 are very small and almost negligible

0.1 to 0.3 are small 0.3 to 0.7 are moderate 0.7 and above are strong

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YX

XY

ssr cov= 陈

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WHAT AFFECTS CORRELATIONS?

4 major factors affect the calculation of correlations:

1. Nonlinearity of relationship 2. Restriction of range (or variance) of X and/or Y http://www.uvm.edu/~dhowell/fundamentals8/

SeeingStatisticsApplets/RangeRestrict.html 3. Use of heterogeneous sub-samples http://www.uvm.edu/~dhowell/fundamentals8/Seei

ngStatisticsApplets/HeteroSubSamp.html 4. Outliers http://www.uvm.edu/~dhowell/fundamentals8/Seei

ngStatisticsApplets/CorrPointRemove.html

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RESTRICTION OF RANGE P205 8/2/2017

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HETEROGENEOUS SUBSAMPLES

A sample is heterogeneous when the observations can be divided one or more distinct sets on the basis of some other variable (e.g. gender, age group, etc.)

Conversely, a homogeneous sample is when the sample is very similar

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HETEROGENEOUS SUBSAMPLES

Let’s take a look at the example from the book. The correlation between height and weight with

both males & females is 0.78—a high correlation…

When we separate out males and females, we get much different results: Males: r = 0.60 Females: r = 0.49

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HETEROGENEOUS SUBSAMPLES 8/2/2017

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OUTLIERS 8/2/2017

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r=.224, p=.509

r=.784, p=.007

CAUTION

CORRELATION ≠

CAUSATION

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CORRELATION ≠ CAUSATION

An observed association between variables does NOT indicate that one variable causes the other—though often we would like this to be the case.

Otts (2005) presents an excellent list of

possible explanations of a significant correlation between two variables. Howell p209

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PREREQUISITES TO ESTABLISH CAUSALITY 1. In order for A to cause B, A must occur before

or at the same time of B. 2. Other variables are ruled out. No other

variables can count for the difference. Randomization is the key to start with.

3. A reasonable explanation has to be provided how A causes B. I we can’t explain why we found what we did, the best that we can say is that there is a relationship between A and B, but more research needs to be carried out to find out why they are related.

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HYPOTHESIS TESTING & CORRELATIONS The rules of hypothesis testing will still apply to

correlations H1: ρ ≠ 0 (two-tailed test) H0: ρ = 0 We set up alpha (a rejection level) and compare. If have strong evidence to reject the null, we say our

correlation is statistically significant.

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HOW DO WE TEST SIGNIFICANCE WITH CORRELATION? What is different?

Degrees of freedom (df) = N – 2 N is the sample size Number of independent pieces of information remaining

after estimating one or more parameters Use the t table in the back of the book Steps:

Calculate r Compute df (N – 2),

Look up critical value in the table using the df

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212

rNrt−

−=

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CORRELATION OUTPUT IN SPSS

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If the p value is <.05, this correlation is statistically significant.

You will see 2 stars if the correlation is significant at alpha level of 0.01, and 1 star at alpha level of .05.

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SUMMARY Correlation is the standardized covariance, can

be positive, negative or 0. 0 means no correlation. The sign of the correlation coefficient reveals the

direction of the relationship. The absolute value of the correlation coefficient

reveals the strength of the relationship. There are different correlation coefficients

appropriate for different situations. We can graph the scatter plot to reveal the

relationship in SPSS. SPSS output provides tests results on the

significance of the correlation coefficients.

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Break

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CHAPTER 10: SIMPLE REGRESSION

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OVERVIEW Independent and dependent variables Regression

The regression line Intercept and slope estimates in a regression line

Unstandardized Standardized

Handling outliers Hypothesis testing Prediction as a purpose SPSS Extension

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INDEPENDENT AND DEPENDENT VARIABLES Independent variables (IV)自变量: Often the

variable manipulated by a researcher. We are usually interested to see the influence of them on something else. We will call them the “predicting” variables or “predictors”, usually noted as “X”. Explanatory variable

Dependent variables (DV)因变量: Often the

variable whose value depends on predictors. In the next two chapters we call these the “criterion” variable or “response” variable, usually noted as “Y”. Response variable

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REGRESSION Regression is the process of finding the best

fitting line using the data we have. This line is most often used to predict or

estimate a score of one variable (Y) from another (X). Predict for future occurrences

Once the relationship between GPA and College Entrance Examination (CEE) is established, we could predict an applicant’s possible GPA based on his CEE scores to help with admission decisions.

Estimate a mean value Once the relationship between height and shoesize is

decided, we can estimate a person’s height based on the footprint.

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VISUAL HELP The regression line: The best fitting line is the

one that minimizes the sum of squared residuals or SSerror , we can give it a notation of .

Our approach of finding the best line is called “least squares regression”.

Draw on the board. No matter how the line tilts, it will always pass one

point, that is

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),( YX

∑ − 2)ˆ( YY

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THE REGRESSION LINE Think back to your high school math class…the

equation for a straight line is:

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Predicted value of Y

Slope of regression

line

Value of X

Intercept

bXaY +=ˆ

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A AND B DEFINE THE LINE Find the line is the same as finding a and b that

defines the line.

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THE REGRESSION LINE

The slope of the best fitting line we just described can be estimated as below, this can be proven, but we will not do that here.

==

X

Y

X

XY

ssr

sb 2

cov

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bXaY +=ˆ

Pearson correlation between X and Y.

Standard deviation of Y.

Standard deviation of X.

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THE REGRESSION LINE: CAN SKIP

After we solve for the slope (b), we can find the intercept a with:

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NXbY

XbYa ∑ ∑−=−=

bXaY +=ˆ

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HYPOTHESIS TESTING

Regression has two coefficients (slope & intercept) Like in correlation, we are testing to see if these values

are different from zero, although we are not interested in testing the intercept (we do not care whether the intercept is 0 or not).

We still use t-test to test this. In SPSS, the t-test has a p-value attached to it. As

usual, if p<alpha(usually 0.05), we reject the null hypothesis and conclude the coefficient is significantly different from 0.

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JUST FOR YOUR INFORMATION t-test for correlation r. t-test for slope coefficient b. It follows a t

distribution with n-2 degrees of freedom.

You can use either equation to double-check the t statistic in the following example

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[ ]21

22

−=−

−== ndf

rNr

srtr

21)1(

1)(1)(

12ˆˆ

−−

−

−=

−=

−

=−−

NNrs

Nsbs

Nsb

Nss

bt

Y

X

YY

X

X

YY

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SPSS DEMO

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Tab10.1 on line. Predicting symptom scores from level of stress. Can you write the equation for this best fitting line?

Intercept

Slope

Standardized slope coefficient. =r (simple regression only.)

Does not include 0. Yes, significant.

Smaller than .05. Significant. Reject null hypothesis and conclude the slope is different from 0.

We don’t care about this.

Unstandardized slope coefficient

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STANDARDIZED DATA So what happens if our data are standardized—that is,

both X and Y are z-scores? The mean of X in this case is ____. The mean of Y in this case is also ____. The standard deviation of X and Y are both _____.

Think now about the equations we used to estimate our regression coefficient—and plug in the values above ( )0 0 0a Y bX b= − = − =

11

Y

X

sb r r r

s = = =

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XbXbXaY β+=+=+= 00ˆ

SPSS call this beta, standardized coefficient.

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PREDICTABLE VARIABILITY & R2

If a correlation (r) is found to be 0.8, we calculate r2 = 0.82 = 0.64

The higher the r2, the better the predictors are working (the more variance in Y that are explained by the predictors).

How do we interpret this? This means that 64% of the variance of Y can be

explained by the variability of X. Remember, this does NOT mean that 64% of Y is

caused by X.

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USE SPSS Sleep and mood: This is fictional data Create scatter plot to check the linear

relationship assumption. Check for outliers.

Delete outliers. Read the output. Make your conclusion. Interpret the results. Evaluate model fit.

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CHAPTER 11: MULTIPLE REGRESSION

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A QUICK EXAMPLE

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CORRELATION

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REGRESSION: ENTER

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IMPROVE THE REGRESSION: STEPWISE

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