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Association and Prediction Using Correlation and Regression. Learning Objectives. Review information from Lecture 10. Understand the relationship between two interval/ratio variables using. Test for association between two variables using correlation and interpret the correlation coefficients. ID: 461445

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Slide1

Slide75

Slide76

SPSS Session 4:

Association and Prediction Using Correlation and Regression

Slide2Learning Objectives

Review information from Lecture 10

Understand the relationship between two interval/ratio variables using

Test for association between two variables using correlation and interpret the correlation coefficients

Using regression, describe how one variable can be used to predict the score in another

Conduct correlation and

regression analyses

in SPSS and interpret the statistical findings

Slide3Review of Lecture 10

Completion of this session

enabled

you to :

Understand

how multiple variables may interact with one another

Appreciate

the role of intervening variables

Be

aware of how interpretation of statistics may be affected by outliers and misinterpretations

Slide4Association Between Variables

Correlation is a statistical test that allows us to gauge the association between two interval/ratio variables.

For example, we would expect age and height to be correlated. As age increases, we expect a similar increase in height.

“Pearson’s R” statistic is the most common correlation test.

Correlation is best understood through the use of a chart called a scatterplot.

Slide5Correlation and Pearson’s r

Pearson’s

r

is the most common correlation coefficient.

It is used to statistically show the

magnitude

and

direction

of a relationship between two variables.

It is on a scale of -1 to 1.

Distance either direction from 0 is crucial and shows

magnitude

.

The sign of the

r

(+/-) shows the

direction

.

Either negative or positive direction

Slide6Scatterplots

Scatterplots produce an useful visualization of the association between two variables.

The independent variable is shown on the horizontal axis (X axis).

The dependent variable is shown on the vertical axis (Y axis).

In the next example, we wanted to describe the

relationship between the age of the person responding to the questionnaire

in our child protection study and

the age of the child in their care.

Slide7In this example of a scatterplot, age of the respondent is on the X axis.Age of the child is on the Y axis.

Slide8Each dot is a single family and represents the point at which the ages of the respondent and child intersect based on the two ages.

Example of a case:

Parent age = 45 yearsChild age = 5 years

Slide9Correlation Lines

Based on the scatterplot, think of a line that could be drawn to represent the relationship between the age of the person responding to the questionnaire and the age of the child in their care.

This line should attempt to minimize the vertical distance between any given point and the line.

It’s often called “the line of best fit”.

Slide10Correlation Line?

Slide11Correlation Line Shown

Slide12The line predicts some of the cases and their association between the ages of the respondent and child very well! These cases sit right on the line!

Slide13The line does not other cases and their ages quite as well. These cases are vertically very far from the line.

Perhaps these were cases where the children were placed in the care of their grandparents after the children were removed from their parents.

Slide14Correlation and Pearson’s r

There are three critical characteristics of correlation needed to properly describe the association between to variables.

MAGNITUDE

DIRECTION

STATISTICAL SIGNIFICANCE

Slide15Magnitude of the Correlation

Distance

either direction from 0 is crucial and shows

magnitude

.

Correlation scores farther away from 0, closer to either -1 or 1, are deemed as stronger.

We would say that correlations of -1 or 1 are perfectly correlated.

Slide16Direction of Correlation

Correlation scores that are above 0 are called positive correlations.

As values for one variable increase, we would expect an associated increase in the other.

Correlation scores that are below 0 are called negative correlations.

As values for one variable increase, we would expect as associated decrease in the other.

Slide17The correlation between the ages of the children and the respondents to the questionnaire in the child protection study was r=.514. The magnitude was moderate as the correlation coefficient was halfway between 0 and 1. Because the correlation score was above 0, we would say that it was a positive correlation.

Slide18Correlation Example 1: GHQ and WAI

We wanted to test for the association between two variables in our child protection study.

The General Health Questionnaire (GHQ) total score which was a measure of psychological distress reported by the respondent answering the questionnaire.

The Working Alliance Inventory (WAI) total score which is a measure of the quality of the relationship that respondents reported having with their the child protection worker.

We hypothesized that respondents reporting greater distress (GHQ scores) would report having a worse relationship (WAI scores) with their child protection worker.

Slide19Correlation Example 1: GHQ and WAI

Our research hypothesis is that GHQ scores and WAI scores are negatively and significantly correlated.

We expected that the

r

correlation coefficient would be less than 0, closer to -1, and statistically significant.

Our

null hypothesis

would be that the two variables would not be significantly associated and thus would have a

r

correlation coefficient

not significantly different from to 0.

Slide20Correlation Example 1: GHQ and WAI

In SPSS, we select the “Analyze” menu, then “Correlate”, and select “Bivariate”.

Slide21Correlation Example 1: GHQ and WAI

The “Bivariate Correlations” window will appear

Find the “

WAI_Total

” and “

GHQ_TotalScore

” variables and add them to the “Variables” list

All the options below which are selected are the usual default.

Slide22Correlation Example 1: GHQ and WAI

Click “OK” to conduct the analysis.

Slide23Correlation Results 1: GHQ and WAI

The results from the analysis indicate that the GHQ and WAI scores have a weak, negative correlation (r= -.184). However, the p-value for this correlation is above the significance level standard of α=.05.The obtained p-value is .075 which is to say the correlation was likely to have happened by chance and is not a significant relationship (p>.05).We would then fail to reject the null hypothesis and say that these two variables are unrelated.

Slide24Correlation Results 1: GHQ and WAI

Slide25Correlation Results 1: GHQ and WAI

Slide26Correlation Example 2: Family Environment

In the child protection study, we have three measures of the characteristics of the family environment using the Family Environment Scale:

FES – Cohesion:

Measure of the perceived level of commitment and support expressed by family members

FES – Expressiveness:

Measure of the degree of emotional openness and encouragement in the family

FES – Conflict:

Measure of familial conflict and expressed anger

Slide27Correlation Example 2: Family Environment

Based on the cohesion, expressiveness, and conflict within a family environment, we can begin to hypothesize about the relationships between the three measures.

We would expect the Cohesion and Expressiveness scores to be positively, strongly, and significantly correlated (correlation coefficient closer to 1).

We would expect that the Conflict scores to be negatively, strongly, and significantly correlated with the Cohesion and Expressiveness scores (correlation coefficient closer to -1).

Our null hypothesis for each of these analyses would be that no score is correlated with any other score and would produce a correlation coefficient not significantly different from 0.

Slide28Correlation Example 2: Family Environment

In SPSS, we select the “Analyze” menu, then “Correlate”, and select “Bivariate”.

Slide29Correlation Example 2: Family Environment

The “Bivariate Correlations” window will appear

Find the

“

FES_Cohesive

”, “

FES_Express

”, and “

FES_Conflict

” variables

and add them to the “Variables” list

All the options below which are selected are the usual default.

Slide30Correlation Example 2: Family Environment

Click “OK” to conduct the analysis.

Slide31Correlation Results 2: Family Environment

Here are the results

Slide32Correlation Results 2: Family Environment

Cohesion and Expressiveness are moderately, positively, and significantly correlated (

r

= .556,

p

<.05).

We can reject our

null hypothesis

that these variables were not associated.

In our study, it appears that there is a moderate and significant between parent or carer reports of the level of

commitment and support expressed by family

members and their degree of emotional openness and encouragement of each other.

Slide33Correlation Results 2: Family Environment

Slide34Correlation Results 2: Family Environment

The degree of family Conflict is moderately, negatively, and significantly correlated with both Cohesion (

r

=

-.486

,

p

<.05

) and Expressiveness

(

r

=

-.403,

p

<.05

).

We

can reject our

null hypothesis

that these variables were not associated.

In

our study, it appears that

increased reports of family conflict is associated with decreased reports of both their level

of commitment and support expressed by family members and their degree of emotional openness and encouragement of each other.

Slide35Moving from Association to Prediction

Moving from Correlation to Regression

Slide36Regression

Regression is an extension of correlation where we take the value of an independent variable and attempt to predict the value in another variable.

Both variables must be interval/ratio level of measurement

Slide37Regression Equation

The equation for regression using one independent variable and one dependent variable is the following:Y is the dependent variable, or the outcome we are trying to predictX1 is independent variable, or the variable we are using to predict the value of the dependent variable (outcome)1 is slope or the size and direction of the relationship between X1 and Y0 is the intercept, or the value of Y when X1 is equal to 0. e is the error term, or how far our prediction is off because we can never perfectly predict a variable using another variable

Slide38

Regression Equation and Lines

X

1 = Predictor – IV

Y = Outcome - DV

1 Slope, or Change in Y for every one unit change in X1

Slide39

Regression Example 1: Age and FES

We will conduct three separate regression analyses in this example.

In each case, we will use age of the child (IV) to predict one of the three FES scores (DV).

FES

– Cohesion:

Measure of the perceived level of commitment and support expressed by family members

Slide40Regression Example 1: Age and FES

Within our child protection study, we wanted to determine if age of the child could predict characteristics of the family environment as reported by the parent or carer responding to the questionnaire.

We would expect that older children are associated with more challenges in the family environment (research hypothesis).

Like correlation, regression uses two interval/ratio variables.

For this analysis, our interval/ratio variables are age of the child and one of the three FES scores.

Slide41Regression Example 1: Age and FES

Our null hypothesis for each analysis is that age of the child does not significantly predict the FES score.In other words, the null hypothesis is that there will be no associated change in the FES score based on a change in age of the child.Statistically, the null hypothesis would indicate that 1 = 0. Recall the regrssion formula:

Slide42

Regression Example 1: Age and FES

Our formula for these analyses is the following:Y is the FES scores, our outcome we are trying to predictX1 is age of the child, our independent variable, or the variable we are using to predict the FES score1 is associated change in FES score for each change in the age of the child0 is the intercept, or the value of a FES score (Y) when the age of the child () is equal to 0. e is the error term, or how far our prediction is off because we can never perfectly predict a variable using another variable

Slide43

Regression Example 1: Age and FES

To conduct each analysis, we need first to select the FES score using the Linear Regression menu.

Select “Analyze”, then “Regression”, then “Linear”.

The Linear Regression window will appear.

Slide44Regression Example 1: Age and FES

Slide45Regression Example 1: Age and FES

The Linear Regression window:

Slide46Find our first dependent variable which will be “FES_Cohesive”Add it to the “Dependent” list.

Slide47Our independent variable is the age of the child.Find the “Child_Age_Yrs” variable and add it to the “Independent(s)” variables list.

Slide48Regression Example 1: Age and FES

Under the “Statistics” menu on the right side of the “Linear Regression” window, select the following:

Regression Coefficients – Estimates: this provides the correlation coefficient

r

-value

for the association between the IV and DV

Model Fit: this provides a value to estimate the percentage of the DV that is explained by the IV

Descriptives: this provides the descriptive statistics for the values in the analysis

Click “Continue”

Click “OK” to conduct the analysis

Slide49Regression Example 1: Age and FES

Slide50Regression Results 1: Age and FES

The first table provides the descriptive statistics for the IV and DV.

Slide51Regression Results 1: Age and FES

The second table offers the correlation coefficients between the age of the child (IV) and the FES – Cohesion scores (DV).From this table, we see that these two variables are significantly associated and have a weak, negative correlation (r = -.244, p<.05).

Slide52Regression Results 1: Age and FES

Produced by the “Model Fit” option, this table provides a summary of the value of our regression equation in predicting FES – Cohesion by using age of the child as our predictor.

Slide53Regression Results 1: Age and FES

We see “R” is our correlation coefficient’s distance from r = 0.“R Square” is r2 or squaring the correlation coefficient.r2 can be interpreted as a percentage of the variance in the DV that is explained by the IV.In this case, age of the child can statistically explain 5.9% of the variation in the FES – Cohesion scores.

Slide54Regression Results 1: Age and FES

Regression tests use the same class as test as the ANOVA, which for this analysis, is below:

Slide55Regression Results 1: Age and FES

The table indicates that from our regression model, we have significantly predicted the FES – Cohesion scores (F=5.880, df= 1,93, p<.05).We can reject our null hypothesis that the age of the child does not predict FES – Cohesion scores.

Slide56Regression Results 1: Age and FES

From the last table, we can construct our regression equation.

Slide57

Regression Results 1: Age and FES

FES-Cohesion =

Slide58

Regression Results 1: Age and FES

From our equation, we can see that for every year that a child is older, there is an average decrease in the FES-Cohesion score of 1.34.The range of FES-Cohesion scores was from 0-9. A decrease of 1.34 on a scale from 0-9 for every year that child is older is a significant and meaningful decrease in cohesion of a family environment as reported by the parent or carer!

Slide59

The regression model significant predicts FES-Cohesion scores

(F=5.880, df= 1,93, p<.05).Age of the child is a significant predictor (t=-2.425, p<.05) of FES-Cohesion. Age of the child explains 5.9% of the variance in the FES-Cohesion scores.

Slide60Regression Example 2: Age and SDQ

To conduct each analysis, we need first to select the

SDQ score

using the Linear Regression menu.

Select “Analyze”, then “Regression”, then “Linear”.

The Linear Regression window will appear.

Slide61Regression Example 2: Age and SDQ scores

We found from the previous analysis that older children in the home are associated with greater difficulties with the cohesion of the family environment.

We wanted to explore this aspect of the family further.

The Strength and Difficulties measure (SDQ) is provides a view of the psychosocial problems of a child as reported by the parent or carer.

We would hypothesize that age of the child would predict increased psychosocial difficulties reported by the parent or carer on the SDQ measure (research hypothesis).

Slide62Regression Example 2: Age and SDQ

We chose to use regression to test the ability of the age of the child to predict SDQ difficulty scores.Both are interval/ratio variables.Our null hypothesis for each analysis is that age of the child does not significantly predict the SDQ scores.In other words, the null hypothesis is that there will be no associated change in the SDQ scores based on a change in age of the child.Statistically, the null hypothesis would indicate that 1 = 0. Recall the regrssion formula:

Slide63

Replace “FES-Cohesive” with “SDQ_TotalDif” from the list on the right.“SDQ_TotalDif” is the new dependent variable in this new analysis.

Slide64Regression Example 2: Age and SDQ

Under the “Statistics” menu on the right side of the “Linear Regression” window, select the following:

Regression Coefficients – Estimates: this provides the correlation coefficient

r

-value

for the association between the IV and DV

Model Fit: this provides a value to estimate the percentage of the DV that is explained by the IV

Descriptives: this provides the descriptive statistics for the values in the analysis

Click “Continue”

Click “OK” to conduct the analysis

Slide65Regression Example 2: Age and SDQ

Slide66Regression Results 2: Age and SDQ

The first table gives the descriptive statistics for the variables in the analysis.

Slide67Regression Results 2: Age and SDQ

The second table provides the correlation coefficients between the two variables.

Slide68Age of the child is not correlated with the SDQ total difficulties score (r= .127, p>.05). The variables are not significantly correlated.The weak, positive correlation likely occurred by chance and not representative of an actual relationship between the two variables.

Slide69We see “R” is our correlation coefficient, r = .127“R Square” is r2 or squaring the correlation coefficient.r2 can be interpreted as a percentage of the variance in the DV that is explained by the IV.In this case, age of the child can statistically explain 1.6% of the variation in the SDQ total difficulties score. This is a very small r2 showing how poorly age of the child predicts SDQ scores.

Slide70Regression Results 2: Age and SDQ

Regression tests use the same class as test as the ANOVA, which for this analysis, is below:

Slide71Regression Results 2: Age and SDQ

The table indicates that from our regression model, we have NOT significantly predicted the SDQ total scores (F=1.524, df= 1,93, p>.05).We have failed to reject our null hypothesis that the age of the child does not predict SDQ total scores .

Slide72The regression model does not significant predict SDQ scores (F=1.524, df= 1,93, p>.05).Age of the child is not a significant predictor (t=-1.234, p>.05) of SDQ scores.

Slide73Regression Results 2: Age and SDQ

It is interesting that the family environment was predicted by the age of the child, but the age of the child did not predict the parent/carer reports of the psychosocial difficulties of the child.

From these two analyses, we can see different results of regression models having completed the regression equation for the one regression model where the independent variable (age of child) did significantly predict the dependent variable (reports of cohesion in the family environment).

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