Download Chapter14 - Karen A. Donahue, Ph.D.

Chapter 15
Introduction
 When referring to interval-ratio variables a commonly
used synonym for association is correlation
 We will be looking for the existence, strength, and
direction of the relationship
 We will only look at bivariate relationships in this
chapter
Scattergrams
 The first step is to construct and examine a
scattergram
 Example in the book
 Analysis of how dual wage-earner families cope with
housework
 They want to know if the number of children in the
family is related to the amount of time the husband
contributes to housekeeping chores
Scattergram of Relationship
Between the Two Variables
 Regression of
Husband’s Hours of
Housework
Hours Per Week Husband Spends on Housework
8
6
 By The Number of
Children in the Family
4
2
0
-2
0
1
Number of Children
2
3
4
5
6
Construction of a Scattergram
 Draw two axes of about equal length and at right angles
to each other
 Put the independent (X) variable along the horizontal
axis (the abscissa) and the dependent (Y) variable along
the vertical axis (the ordinate)
 For each person, locate the point along the abscissa that
corresponds to the scores of that person on the X
variable
 Draw a straight line up from that point and at right angles to the
axis
 Then locate the point along the ordinate that corresponds to
the score of that same case on the Y variable
 Place a dot there to represent the case, and then repeat with all
cases
Regression Line and its Purpose
 It checks for linearity of the data points on the
scattergram
 It gives information about the existence, strength, and
direction of the association
 It is used to predict the score of a case on one variable
from the score of that case on the other variable
 It is a floating mean through all the data points
Scattergram of Relationship
Between the Two Variables
 Regression of
Husband’s Hours of
Housework
Hours Per Week Husband Spends on Housework
8
6
 By The Number of
Children in the Family
4
2
0
-2
0
1
Number of Children
2
3
4
5
6
Existence of a Relationship
 Two variables are associated if the distributions of Y
change for the various conditions of X
 The scores along the abscissa (number of children)
are conditions of values of X
 The dots above each X value can be thought of as the
conditional distributions of Y (scores on Y for each
value of X)

In other words, Y tends to increase as X increases
Existence of a Relationship
 The existence of a relationship is reinforced by the fact
that the regression line lies at an angle to the X axis
(the abscissa)
 There is no linear relationship between two interval-
level variables when the regression line on a scattergram
is parallel to the horizontal axis
Scattergram of Relationship
Between the Two Variables
 Regression of
Husband’s Hours of
Housework
Hours Per Week Husband Spends on Housework
8
6
 By The Number of
Children in the Family
4
2
0
-2
0
1
Number of Children
2
3
4
5
6
Strength of the Association
 The strength of the association is judged by
observing the spread of the dots around the
regression line
 A perfect association between variables can be seen on a
scattergram when all dots lie on the regression line
 The closer the dots to the regression line, the stronger
the association
 So, for a given X. there should not be much variety on
the Y variable
Scattergram of Relationship
Between the Two Variables
 Regression of
Husband’s Hours of
Housework
Hours Per Week Husband Spends on Housework
8
6
 By The Number of
Children in the Family
4
2
0
-2
0
1
Number of Children
2
3
4
5
6
Direction of the Relationship
 The direction of the relationship can be judged
by observing the angle of the regression line
with respect to the abscissa
 The relationship is positive when the line slopes
upward from left to right
 The association is negative when it slopes down
 Your book shows a positive relationship, because
cases with high scores on X also tend to have high
scores on Y
 For a negative relationship, high scores on X would
tend to have low scores on Y, and vice versa
 Your book also shows a zero relationship—no
association between variables, in that they are
randomly associated with each other
Linearity
 The key assumption (first step in the five-step
model) with correlation and regression is that the
two variables have an essentially linear
relationship
 The points or dots must form a pattern of a straight line
 It is important to begin with a scattergram before doing
correlations and regressions
 If the relationship is nonlinear, you may need to treat
the variables as if they were ordinal rather than intervalratio
Regression and Prediction
 The final use of the scattergram is to predict scores
of cases on one variable from their score on the
other
 May want to predict the number of hours of
housework a husband with a family of four
children would do each week
 You use regression to predict outside the range of
the data with caution, since you do not have any
data points to show what happens beyond the
scope of the data—it may have suddenly gone
down
The Predicted Score on Y
 The symbol for this is Y’, or Y prime, though in other




books, it is most often Y hat, but that symbol is difficult
to do on a computer or to print in books
It is found by first locating the score on X (X=4, for four
children) and then drawing a straight line from that
point on the abscissa to the regression line
From the regression line, another straight line parallel
to the abscissa is drawn across to the Y axis or ordinate
Y’ is found at the point where the line from the
regression line crosses the Y axis
Or, you can compute Y’ = a + bX
 Y’ is the expected Y value for a given X
Formula for the Regression Line
 The formula for a straight line that fits closest to the
conditional means of Y





Y = a + bX
Where Y = score on the dependent variable
a = the Y intercept or the point where the regression line
crosses the Y axis
b = the slope of the regression line or the amount of change
produced in Y by a unit change in X
X = score on the independent variable
Regression Line
 The position of the least-squares regression line is
defined by two elements
 The Y intercept and the slope of the line
 It also crosses the point where the mean of X meets the
mean of Y
 The weaker the effect of X on Y (the weaker the
association between the variables) the lower the
value of the slope (b)
 If the two variables are unrelated, the least-squares
regression line would be parallel to the abscissa,
and b would be 0 (the line would have no slope)
Scattergram of Relationship
Between the Two Variables
 Regression of
Husband’s Hours of
Housework
Hours Per Week Husband Spends on Housework
8
6
 By The Number of
Children in the Family
4
2
0
-2
0
1
Number of Children
2
3
4
5
6
Equations for the Slope of the
Regression Line
 You need to compute “b” first, since it is needed in
the formula for “a”
 Slope:
 X  X Y  Y 
b
2
 X  X 
 Which is the covariance of X and Y divided by the
variance of X
Interpretation of the Value of the Slope
 If you put your scattergram on graph paper, you
can see that as X increases one box, “b” is how
many units that Y increases on the regression
line
 So, a slope of .69 indicates that, for each unit
increase in X, there is an increase of .69 units in
Y
 If the slope is 1.5, for every unit of change in X, there
is an increase of 1.5 units in Y
 They refer to units, since correlation and regression
allow you to compare apples and oranges—two
completely different variables
Scattergram of Relationship
Between the Two Variables
 Regression of
Husband’s Hours of
Housework
Hours Per Week Husband Spends on Housework
8
6
 By The Number of
Children in the Family
4
2
0
-2
0
1
Number of Children
2
3
4
5
6
Interpretation of “b” cont.
 So, to find what one unit of X is or one unit of Y is,
you have to go back to the labels for each variable
 For the example in your book which has a “b”
(beta) of .69
 The addition of each child (an increase of one unit in
X—one unit is one child)
 Results in an increase of .69 hours of housework
being done by the husband (an increase of .69
units—or hours—in Y)
Formula for the Intercept of the Regression
Line
a  Y  bX
Interpretation of the Intercept
 The intercept for the example in the book is 1.49
 The least-squares regression line will cross the Y
axis at the point where Y equals 1.49
 You need a second point to draw the regression
line
 You can begin at Y of 1.49, and for the next value of X,
which is 1 child, you will go up .69 units of Y
 Or, you can use the intersection of the mean of X and
the mean of Y—the regression line always goes through
this point
Interpretation of “a” cont.
 Most of the time, you can’t interpret the value of
the intercept
 Technically, it is the value that Y would take if X were
zero




But, most often a zero X is not meaningful
Or, in the case in your book, zero is outside the range of the data
You don’t have any information about the hours of housework
that husbands do when they have no children
Technically, the intercept of 1.49 is the amount of predicted
housework a husband with zero children would do, but you can’t
say that with certainty
Least Squares Regression Line
 Now that you know “a” and “b”, you can fill in
the full least-squares regression line
 Y = a + bX
 Y = (1.49) + (.69) X
 This formula can be used to predict scores on Y as
was mentioned earlier


For any value of X, it will give you the predicted value of
Y (Y’)
The predictions of husband’s housework are “educated
guesses”
 The accuracy of our predictions will increase as
relationships become stronger (as dots are closer to
the regression line)
The Correlation Coefficient (Pearson’s
r)
 Pearson’s r varies from 0 to plus or minus 1
 With 0 indicating no association
 And + 1 and – 1 indicating perfect positive and perfect
negative relationships
 The definitional formula for Pearson’s r is in your book
 Similar to the formula for b (beta), the numerator is the
covariation between X and Y (usually called the
covariance)
Interpretating r and r-squared
 Interpretation of “r” will be the same as all the
other measures of association
 An “r” of .5 would be a moderate positive linear
relationship between the variables
 Interpretation of the Coefficient of
Determination (r-squared)
 The square of Pearson’s r is also called the
coefficient of determination
 While “r” measures the strength of the linear
relationship between two variables

But values between 0 and 1 or -1 have no direct interpretation
Interpretation, cont.
 The coefficient of determination can be interpreted
with the logic of PRE (proportional reduction in error)
 First Y is predicted while ignoring the information supplied by
X
 Second the independent variable is taken into account when
predicting the dependent
 When working with variables measured at the interval-
ratio level, the predictions of Y under the first condition
(while ignoring X) will be the mean of the Y scores (Y
bar) for every case
 We know that the mean of any distribution is closer than any
other point to all the scores in the distribution
Interpretation, cont.
 Will make many errors in predicting Y
 The amount of error is shown in Figure 16.6
 The formula for the error is the sum of (Y minus Y
bar) squared
 This is called the total variation in Y, meaning the
total amount that all the points are off the mean of Y
 The next step will be to find the extent to
which knowledge of X improves our ability to
predict Y (Will we make predictions that come
closer to the actual points than predictions
made using the mean of Y?)
Interpretation, cont.
 If the two variables have a linear relationship, then
predicting scores on Y from the least-squares
regression equation will use knowledge of X and
reduce our errors of prediction
 The formula for the predicted Y score for each value of
X will be: Y’ = a + bX
 This is also the formula for the regression line
Unexplained Variation
 That suggests that some of the variation in Y is
unexplained by X
 The proportion of the total variation in Y unexplained
by X can also be found by subtracting the value of rsquared from 1.00
 Other variables will be needed to explain one hundred
percent of the variation in Y (the dependent variable)
Unexplained Variation, cont.
 Unexplained variation is usually attributed to the
influence of three things:
 Some combination of other variables, as in the
example of the husband’s housework
 Measurement error

People over or under estimate how much time they spend
doing housework
 Random chance

Your sample may be biased, particularly if it is small
Testing Pearson’s r for Significance
 When “r” is based on data from a random sample,
you need to test “r” for its statistical significance
 When testing Pearson’s r for significance, the null
hypothesis is that there is no linear association
between the variables in the population from
which the sample was drawn
 We will use the t distribution for this test
Assumptions for the Significance Test
 We make some additional assumptions in Step 1
 Need to assume that both variables are normal in
distribution
 Need to assume that the relationship between the
two variables is roughly linear in form
 The third new assumption involves the concept of
homoscedasticity
Homoscedasticity
 A homoscedastistic relationship is one where the
variance of the Y scores is uniform for all values of X
 If the Y scores are evenly spread above and below the regression
line for the entire length of the line, the relationship is
homoscedastistic
 If the variance around the regression line is greater at one end
or the other, the relationship is heteroscedastistic
 A visual inspection of the scattergram is usually sufficient to
find the extent the relationship conforms to the assumptions of
linearity and homoscedasticity
 If the data points fall in a roughly symmetrical, cigar-shaped
pattern, whose shape can be approximated with a straight line,
then it is appropriate to proceed with this test of significance
Scattergram of Relationship
Between the Two Variables
 Regression of
Husband’s Hours of
Housework
Hours Per Week Husband Spends on Housework
8
6
 By The Number of
Children in the Family
4
2
0
-2
0
1
Number of Children
2
3
4
5
6