Download Chapter 8 Review, Part 1

The Regression Equation
Using the regression equation to
individualize prediction and move beyond
saying that everyone is equal, that
everyone should score right at the mean
Best fitting line: A review
The definition of the best fitting line
plotted on t axes
• A “best fitting line” minimizes the average
squared vertical distance of Y scores in the sample
(expressed as tY scores) from the line.
• The best fitting line is a least squares, unbiased
estimate of values of Y in the sample.
• The generic formula for a line is Y=mx+b where
m is the slope and b is the Y intercept.
• Thus, any specific line, such as the best fitting
line, can be defined by its slope and its intercept.
The intercept of the best fitting line
plotted on t axes
The origin is the point where both tX and
tY=0.000
• So the origin represents the mean of both
the X and Y variable
• When plotted on t axes all best fitting lines
go through the origin.
• Thus, the tY intercept of the best fitting line =
0.000.
The slope of and formula for the
best fitting line
• When plotted on t axes the slope of the
best fitting line = r, the correlation
coefficient.
• To define a line we need its slope and Y
intercept
• r = the slope and tY intercept=0.00
• The formula for the best fitting line is
therefore tY=rtX + 0.00 or tY= rtX
Here’s how a visual representation of the best fitting line
(slope = r, Y intercept = 0.000) and the dots representing
tX and tY scores might be described. (Whether the
correlation is positive of negative doesn’t matter.)
• Perfect - scores fall exactly on a straight line.
• Strong - most scores fall near the line.
• Moderate - some are near the line, some not.
• Weak - the scores are only mildly linear.
• Independent - the scores are not linear at all.
Strength of a relationship
1.5
Perfect
1.0
0.5
0
-1.5
-1.0
-0.5
0
0.5
-0.5
-1.0
-1.5
1.0
1.5
Strength of a relationship
1.5
1.0
Strong
r about .800
0.5
0
-1.5
-1.0
-0.5
0
0.5
-0.5
-1.0
-1.5
1.0
1.5
Strength of a relationship
1.5
1.0
0.5
0
-1.5
-1.0
-0.5
0
0.5
-0.5
-1.0
-1.5
1.0
1.5
Moderate
r about .500
Strength of a relationship
r about 0.000
1.5
1.0
0.5
0
-1.5
-1.0
-0.5
0
0.5
1.0
1.5
-0.5
-1.0
Independent
-1.5
r=.800, the formula for the best
fitting line = ???
1.5
1.0
0.5
0
-1.5
-1.0
-0.5
0
0.5
-0.5
-1.0
-1.5
1.0
1.5
r=-.800, the formula for the best
fitting line = ???
1.5
1.0
0.5
0
-1.5
-1.0
-0.5
0
0.5
-0.5
-1.0
-1.5
1.0
1.5
r=0.000, the formula for the best
fitting line is:
1.5
1.0
0.5
0
-1.5
-1.0
-0.5
0
0.5
-0.5
-1.0
-1.5
1.0
1.5
Notice what that formula for
independent variables says
• tY = rtX = 0.000 (tX) = 0.000
• When tY = 0.000, you are at the mean of Y
• So, when variables are independent, the best
fitting line says that the best estimate of Y scores
in the sample is back to the mean of Y regardless
of your score on X
• Thus, when variables are independent we go back
to saying everyone will score right at the mean
A note of caution: Watch out for the plot
for which the best fitting line is a curve.
1.5
1.0
0.5
0
-1.5
-1.0
-0.5
0
0.5
-0.5
-1.0
-1.5
1.0
1.5
Moving from the best fitting line
to the regression equation and the
regression line.
The best fitting line (tY=rtX) was
the line closest to the Y values in
the sample.
But what should we do if we
want to go beyond our sample
and use a version of our best
fitting line to make
individualized predictions for the
rest of the population?
What do we need to do to be able
to use the regression equation
• tY' = rtX
• Notice this is not quite the same as the formula for
the best fitting line. The formula now reads tY'
(read t-y-prime). Not tY.
• tY' is the predicted score on the Y variable for
every X score in the population within the range
observed in our random sample.
• Before we were describing the linear relationship
of the X and Y variable in our sample. Now we are
predicting estimated Z scores (t scores) for most or
all of the population.
This is one of the key points in the course;
a point when things change radically.
Up to this point, we have just been describing
scores, means and relationships. We have not
yet gone beyond predicting that everyone in the
population who was not in our sample will
score at the mean of Y.
But now we want to be able to make
individualized predictions for the rest of the
population, the people not in our sample and
for whom we don’t have Y scores
That’s dangerous. Our first rule
as scientists in “Do not increase
error.” Individualizing prediction
can easily do that.
Let me give you an example.
Assume, you are the personnel
officer for a mid size company.
•
•
•
•
You need to hire a typist.
There are 2 applicants for the job.
You give the applicants a typing test.
Which would you hire: someone who types
6 words a minute with 12 mistakes or
someone who types 100 words a minute
with 1 mistake.
Who would you hire?
• Of course, you would predict that the second
person will be a better typist and hire that person.
• Notice that we never gave the person with 6
words/minute a chance to be a typist in our firm.
• We prejudged her on the basis of the typing test.
• That is probably valid in this case – a typing test
probably predicts fairly well how good a typist
someone will be.
But say the situation is a little more complicated!
• You have several applicants for a leadership
position in your firm.
• But it is not 2002, it is 1957, when we knew that
only white males were capable of leadership in
corporate America.
• That is, we all “know” that leadership ability is
correlated with both gender and skin color, white
and male are associated with high leadership
ability and darker skin color and female gender
with lower leadership ability.
• We now know this is absurd, but lots of people
were never given a chance to try their hand at
leadership, because of pre-judgement that you can
now see as obvious prejudice.
We would have been much better off saying
that everyone is equal, everyone should be
predicted to score at the mean.
• Pre-judgements on the basis of supposed
relationships between variables that have no real
scientific support are a form of prejudice.
• They cost potential leaders jobs in which they
could have shown their ability. That is unfair.
• Moreover, by excluding such individuals, you
narrow the talent pool of potential leaders. The
more restricted the group of potential leaders, the
less talented the average leader will be.
• This is why aristocracies don’t work in the long
run. The talent pool is too small.
So, to avoid prejudice you must start with the
notion that everyone will score at the mean.
In correlational language, to make that
prediction you have to hypothesize that
rho = 0.000.
Only if you can disprove the notion that
rho = 0.000 and no other time, should you make
any other prediction
We call the hypothesis that
rh0=0.000 the null hypothesis
The symbol for the null hypothesis is
H0. We will see the null hypothesis
many times during the rest of this
course. It is the hypothesis that you
will learn to test statistically.
Confidence intervals around rhoT
Confidence intervals around rhoT
– relation to Chapter 6
• In Chapter 6 we learned to create confidence intervals around muT that
allowed us to test a theory.
• To test our theory about mu we took a random sample, computed the
sample mean and standard deviation, and determined whether the
sample mean fell into that interval.
• If the sample mean fell into the confidence interval, there was some
support for our theory, and we held onto it.
• The interesting case was when muT fell outside the confidence interval.
• In that case, the data from our sample falsified our theory, so we had to
discard the theory and the estimate of mu specified by the theory
If we discard a theory based prediction,
what do we use in its place?
• Generally, our best estimate of a population
parameter is the sample statistic that estimates it.
• Our best estimate of mu has been and is the
sample mean, X-bar.
• Since X-bar fell outside the confidence interval,
we discarded our theory.
• Then we were back to using X-bar, the sample
mean that fell outside the confidence interval and
falsified our theory, as our best (least squares,
unbiased, consistent estimate) of mu.
To test any theory about any
population parameter, we go
through similar steps:
• We theorize about the value of the population parameter.
• We obtain some measure of the variability of sample based
estimates of the population parameter.
• We create a test of the theory about the population
parameter by creating a confidence interval, almost always
a CI.95.
• We then obtain and measure the parameter in a random
sample.
The sample statistic will fall inside
or outside of the CI.95
• If the sample statistic falls inside the confidence
interval, our theory has received some support and
we hold on to it.
• But the more interesting case is when the sample
statistic falls outside the confidence interval.
• Then we must discard the theory and the theory
based estimate of the population parameter.
• In that case, our best estimate of the population
parameter is the sample statistic
• Remember, the sample statistic is a least squares,
unbiased, consistent estimate of its population
parameter.
We are going to do the same
thing with a theory about rho
• rho is the correlation coefficient for the
population.
• If we have a theory about rho, we can create a
95% confidence interval into which we expect r
will fall.
• An r computed from a random sample will then
fall inside or outside the confidence interval.
When r falls inside or outside of the
CI.95 around rhoT
• If r falls inside the confidence interval, our theory
about rho has received some support and we hold
on to it.
• But the more interesting case is when r falls
outside the confidence interval.
• Then we must discard the theory and the theory
based estimate of the population parameter.
• In that case, our best estimate of rho is the r we
found in our random sample
• Thus, when r falls outside the CI.95 we can go back
to using it as a least squares unbiased estimate of
rho.
Then what?
• Then we can use the r from our sample, the
r that falsified the theory that rho=0.000, in
the regression equation:
• tY'=rtX
To repeat
• If rho = 0.000, we should go back to saying
everyone is equal, everyone will score at the mean
of Y.
• To be fair and avoid doing damage, we must test
the hypothesis that rho=0.000 before doing
anything else.
• To test the theory that rho=0.00, we create a CI.95
for rho=0.000.
• If, and only if, we disprove the notion that
rho=0.000 by having r fall outside the CI.95 can we
use r in the regression equation, tY'=rtX.
I could teach you how to calculate the
confidence interval for rho=0.000
• But other people have already calculated the
intervals for many different df.
• Those calculations are summarized in the r
table
How the r table is laid out:
the important columns
– Column 1 of the r table shows degrees of freedom for
correlation and regression (dfREG)
– dfREG=nP-2
– Column 2 shows the CI.95 for varying degrees of
freedom
– Column 3 shows the absolute value of the r that falls
just outside the CI.95. Any r this far or further from
0.000 falsifies the hypothesis that rho=0.000 and can be
used in the regression equation to make predictions of
Y scores for people who were not in the original sample
but who were part of the population from which the
sample is drawn.
df
nonsignificant
.05
.01
If r falls in.9999
within the 95% CI
.997
around
0.000,
.950
.990then the result is
.878
not .959
significant.
1
-.996 to .996
2
-.949 to .949
3
-.877 to .877
4
to .810value .811
.917
Does
the-.810
absolute
5
-.753
to .753
.754
.874
Find
your
degrees
of
or exceed
6 r equal
-.706
to .706 the .707
.834
of
freedom
(N
-2)
7value in-.665
.665 p
.666 cannot .798
thistocolumn?
You
reject
in-.631
thistocolumn
8
.631
.632
.765
the
null hypothesis.
9
-.601 to .601
.602
.735
You
can
10
-.575 to .575
.576
.708use it in the
r is significant
with
11
-.552 to .552
.553 regression
.684 equation to
alpha
= .05.
You
must
12
-.531
to .531assume .532
.661 Y scores.
estimate
.
.
.
that rho =. 0.00.
.
.
.
.
.
.
.
.
If r is significant
you
100
-.194 to .194
.195
.254
can
consider
it
an
unbiased,
200
-.137 to .137
.138
.181
300
-.112
to .112 estimate
.113
least
squares
of rho. .148
500
-.087 toalpha
.087 = .05..088
.115
1000
-.061 to .061
.062
.081
2000
-.043 to .043
.044
.058
10000
-.019 to .019
.020
.026
Pizza and horror films
(scale 0-9)
H1: People who enjoy food
with strong flavors also
enjoy other strong
sensations.
H0: There is no relationship
between enjoying food
with strong flavors and
enjoying other strong
sensations.
horror
anchovies films
7
7
7
9
3
8
3
6
0
9
8
6
4
5
1
2
1
1
1
6
Can we reject the null hypothesis?
Can we reject the null hypothesis?
8
6
Pizza
4
2
0
0
2
4
Horror films
6
8
Can we reject the null hypothesis?
We do the math and we find that:
r = .352
df = 8
r table
df
nonsignificant
.05
.01
1
2
3
4
5
6
7
-.996 to .996
-.949 to .949
-.877 to .877
-.810 to .810
-.753 to .753
-.706 to .706
-.665 to .665
-.631 to .631
-.601 to .601
-.575 to .575
-.552 to .552
-.531 to .531
.
.
.
-.194 to .194
-.137 to .137
-.112 to .112
-.087 to .087
-.061 to .061
-.043 to .043
-.019 to .019
.997
.950
.878
.811
.754
.707
.666
.9999
.990
.959
.917
.874
.834
.798
.765
.735
.708
.684
.661
.
.
.
.254
.181
.148
.115
.081
.058
.026
8
9
10
11
12
.
.
.
100
200
300
500
1000
2000
10000
.632
.602
.576
.553
.532
.
.
.
.195
.138
.113
.088
.062
.044
.020
This finding falls within the CI.95
around 0.000
•
•
•
•
•
We call such findings “nonsignificant”
Nonsignificant is abbreviated n.s.
We would report these finding as follows
r (8)=0.352, n.s.
Given that it fell inside the CI.95, we must assume
that rho actually equals zero and that our sample r
is not 0.000 solely because of sampling
fluctuation.
• We go back to predicting that everyone will score
at the mean of Y.
That seems like a good idea:
• I would guess that like most variables,
desire for anchovy pizza and horror movies
are not really correlated.
• This sample probably has an r of .352 solely
because of the way samples of this size
fluctuate around a rho of zero
How to report a signficant r
• For example, let’s say that you had a sample
(nP=30) and r = -.400
• Looking under nP-2=28 dfREG, we find the interval
consistent with the null is between -.360 and
+.360
• So we are outside the CI.95 for rho=0.000
• We would write that result as r(28)=-.400, p<.05
• That tells you the dfREG, the value of r, and that
you can expect an r that far from 0.000 five or
fewer times in 100 when rho = 0.000
Then there is Column 4
• Column 4 shows the values that lie outside a CI.99
• (The CI.99 itself isn’t shown like the CI.95 in Column 2
because it isn’t important enough.)
• However, Column 4 gives you bragging rights.
• If your r is as far or further from 0.000 as the number in
Column 4, you can say there is 1 or fewer chance in 100 of
an r being this far from zero (p<.01).
• For example, let’s say that you had a sample (nP=30) and r
= -.525.
• The critical value at .01 is .463. You are further from 0.00
than that.So you can brag.
• You write that result as r(28)=-.525, p<.01.
To summarize
• If r falls inside the CI.95 around 0.000, it is
nonsignificant (n.s.) and you can’t use the
regression equation (e.g., r(28)=.300, n.s.
• If r falls outside the CI.95, but not as far from 0.000
as the number in Column 4, you have a significant
finding and can use the regression equation (e.g.,
r(28)=-.400,p<.05
• If r is as far or further from zero as the number in
Column 4, you can use the regression equation
and brag while doing it (e.g., r(28)=-.525, p<.01