171
Figure 7.1
Me playing
with my dinga-ling
in the
Holimarine Talent
Show. Note the
groupies queuing
up at the front
7.1.  What will this chapter tell me? 1
Although none of us can know the future, predicting it is so important that organisms are
hard wired to learn about predictable events in their environment. We saw in the previous
chapter that I received a guitar for Christmas when I was 8. My first foray into public
performance was a weekly talent show at a holiday camp called ‘Holimarine’ in Wales (it
doesn’t exist anymore because I am old and this was 1981). I sang a Chuck Berry song called
‘My ding-a-ling’1
and to my absolute amazement I won the competition.2
Suddenly other 8
year olds across the land (well, a ballroom in Wales) worshipped me (I made lots of friends
after the competition). I had tasted success, it tasted like praline chocolate, and so I wanted
to enter the competition in the second week of our holiday. To ensure success, I needed to
know why I had won in the first week. One way to do this would have been to collect data
and to use these data to predict people’s evaluations of children’s performances in the contest
1
It appears that even then I had a passion for lowering the tone of things that should be taken seriously.
2
I have a very grainy video of this performance recorded by my dad’s friend on a video camera the size of a
medium-sized dog that had to be accompanied at all times by a ‘battery pack’ the size and weight of a tank. Maybe
I’ll put it up on the companion website …
7Regression
172 DISCOVERING STATISTICS USING SAS
from certain variables: the age of the performer, what type of performance they gave (singing,
telling a joke, magic tricks), and maybe how cute they looked. A regression analysis on
these data would enable us to predict future evaluations (success in next week’s competition)
based on values of the predictor variables. If, for example, singing was an important factor
in getting a good audience evaluation, then I could sing again the following week; however,
if jokers tended to do better then I could switch to a comedy routine. When I was 8 I wasn’t
the sad geek that I am today, so I didn’t know about regression analysis (nor did I wish to
know); however, my dad thought that success was due to the winning combination of a
cherub-looking 8 year old singing songs that can be interpreted in a filthy way. He wrote me
a song to sing in the competition about the keyboard player in the Holimarine Band ‘messing
about with his organ’, and first place was mine again. There’s no accounting for taste.
7.2.  An introduction to regression 1
In the previous chapter we looked at how to measure relationships between two variables.
These correlations can be very useful but we can take this process a step further and predict
one variable from another. A simple example might be to try to predict levels of stress from
the amount of time until you have to give a talk. You’d expect this to be a negative relationship
(the smaller the amount of time until the talk, the larger the anxiety). We could then
extend this basic relationship to answer a question such as ‘if there’s 10 minutes to go until
someone has to give a talk, how anxious will they be?’ This is the essence of regression
analysis: we fit a model to our data and use it to predict values of the dependent variable
from one or more independent variables.3
Regression analysis is a way of predicting an outcome
variable from one predictor variable (simple regression) or several predictor variables
(multiple regression). This tool is incredibly useful because it allows us to go a step beyond
the data that we collected.
In section 2.4.3 I introduced you to the idea that we can predict any data using the following
general equation:
outcomei = ðmodelÞ + errori (7.1)
This just means that the outcome we’re interested in for a particular person can be predicted
by whatever model we fit to the data plus some kind of error. In regression, the
model we fit is linear, which means that we summarize a data set with a
straight line (think back to Jane Superbrain Box 2.1). As such, the word
‘model’ in the equation above simply gets replaced by ‘things that define
the line that we fit to the data’ (see the next section).
With any data set there are several lines that could be used to summarize
the general trend and so we need a way to decide which of many possible
lines to chose. For the sake of making accurate predictions we want
to fit a model that best describes the data. The simplest way to do this
would be to use your eye to gauge a line that looks as though it summarizes
the data well. You don’t need to be a genius to realize that the ‘eyeball’
method is very subjective and so offers no assurance that the model
is the best one that could have been chosen. Instead, we use a mathematical technique called
the method of least squares to establish the line that best describes the data collected.
3
I want to remind you here of something I discussed in Chapter 1: SAS refers to regression variables as dependent
and independent variables (as in controlled experiments). However, correlational research by its nature seldom
controls the independent variables to measure the effect on a dependent variable and so I will talk about ‘independent
variables’ as predictors, and the ‘dependent variable’ as the outcome.
How do I fit a straight
line to my data?
173CHAPTER 7 Regression
7.2.1.   Some important information about straight lines 1
I mentioned above that in our general equation the word ‘model’ gets replaced by ‘things
that define the line that we fit to the data’. In fact, any straight line can be defined by two
things: (1) the slope (or gradient) of the line (usually denoted by b1
); and (2) the point at
which the line crosses the vertical axis of the graph (known as the intercept of the line, b0
).
In fact, our general model becomes equation (7.2) below in which Yi
is the outcome that
we want to predict and Xi
is the ith participant’s score on the predictor variable.4
Here b1
is
the gradient of the straight line fitted to the data and b0
is the intercept of that line. These
parameters b1
and b0
are known as the regression coefficients and will crop up time and
time again in this book, where you may see them referred to generally as b (without any
subscript) or bi
(meaning the b associated with variable i). There is a residual term, εi
, which
represents the difference between the score predicted by the line for participant i and the
score that participant i actually obtained. The equation is often conceptualized without this
residual term (so ignore it if it’s upsetting you); however, it is worth knowing that this term
represents the fact that our model will not fit the data collected perfectly:
Yi = ðb0 + b1XiÞ + εi (7.2)
A particular line has a specific intercept and gradient. Figure 7.2 shows a set of lines that
have the same intercept but different gradients, and a set of lines that have the same gradient
but different intercepts. Figure 7.2 also illustrates another useful point: the gradient of
the line tells us something about the nature of the relationship being described. In Chapter
6 we saw how relationships can be either positive or negative (and I don’t mean the difference
between getting on well with your girlfriend and arguing all the time!). A line that
has a gradient with a positive value describes a positive relationship, whereas a line with a
negative gradient describes a negative relationship. So, if you look at the graph in Figure 7.2
4
You’ll sometimes see this equation written as:
Yi
= (β0
+ β1
Xi
) + εi
The only difference is that this equation has got βs in it instead of bs and in fact both versions are the same thing,
they just use different letters to represent the coefficients.
Figure 7.2
Lines with the
same gradients
but different
intercepts,
and lines that
share the same
intercept but
have different
gradients
174 DISCOVERING STATISTICS USING SAS
in which the gradients differ but the intercepts are the same, then the dashed line describes a
positive relationship whereas the solid line describes a negative relationship. Basically, then,
the gradient (b1
) tells us what the model looks like (its shape) and the intercept (b0
) tells us
where the model is (its location in geometric space).
If it is possible to describe a line knowing only the gradient and the intercept of that line,
then we can use these values to describe our model (because in linear regression the model
we use is a straight line). So, the model that we fit to our data in linear regression can be
conceptualized as a straight line that can be described mathematically by equation (7.2). With
regression we strive to find the line that best describes the data collected, then estimate the
gradient and intercept of that line. Having defined these values, we can insert different values
of our predictor variable into the model to estimate the value of the outcome variable.
7.2.2.   The method of least squares 1
I have already mentioned that the method of least squares is a way of finding the line that
best fits the data (i.e. finding a line that goes through, or as close to, as many of the data
points as possible). This ‘line of best fit’ is found by ascertaining which line, of all of the
possible lines that could be drawn, results in the least amount of difference between the
observed data points and the line. Figure 7.3 shows that when any line is fitted to a set of
data, there will be small differences between the values predicted by the line and the data
that were actually observed.
Back in Chapter 2 we saw that we could assess the fit of a model (the example we used
was the mean) by looking at the deviations between the model and the actual data collected.
These deviations were the vertical distances between what the model predicted and
each data point that was actually observed. We can do exactly the same to assess the fit of
a regression line (which, like the mean, is a statistical model). So, again we are interested
in the vertical differences between the line and the actual data because the line is our
model: we use it to predict values of Y from values of the X variable. In regression these
differences are usually called residuals rather than deviations, but they are the same thing.
As with the mean, data points fall both above (the model underestimates their value) and
Figure 7.3
This graph
shows a
scatterplot of
some data with a
line representing
the general trend.
The vertical
lines (dotted)
represent the
differences
(or residuals)
between the
line and the
actual data
175CHAPTER 7 Regression
below (the model overestimates their value) the line, yielding both positive and negative
differences. In the discussion of variance in section 2.4.2 I explained that if we sum positive
and negative differences then they cancel each other out and that to circumvent this
problem we square the differences before adding them up. We do the same thing here. The
resulting squared differences provide a gauge of how well a particular line fits the data: if
the squared differences are large, the line is not representative of the data; if the squared
differences are small, the line is representative.
You could, if you were particularly bored, calculate the sum of squared differences (or
SS for short) for every possible line that is fitted to your data and then compare these
‘goodness-of-fit’ measures. The one with the lowest SS is the line of best fit. Fortunately
we don’t have to do this because the method of least squares does it for us: it selects the
line that has the lowest sum of squared differences (i.e. the line that best represents the
observed data). How exactly it does this is by using a mathematical technique for finding
maxima and minima and this technique is used to find the line that minimizes the sum of
squared differences. I don’t really know much more about it than that, to be honest, so I
tend to think of the process as a little bearded wizard called Nephwick the Line Finder who
just magically finds lines of best fit. Yes, he lives inside your computer. The end result is that
Nephwick estimates the value of the slope and intercept of the ‘line of best fit’ for you. We
tend to call this line of best fit a regression line.
7.2.3.  Assessing the goodness of fit: sums of squares,
R and R2 1
Once Nephwick the Line Finder has found the line of best fit it is important that we assess
how well this line fits the actual data (we assess the goodness of fit of the model). We do this
because even though this line is the best one available, it can still be a lousy fit to the data!
In section 2.4.2 we saw that one measure of the adequacy of a model is the sum of squared
differences (or more generally we assess models using equation (7.3) below). If we want to
assess the line of best fit, we need to compare it against something, and the thing we choose
is the most basic model we can find. So we use equation (7.3) to calculate the fit of the most
basic model, and then the fit of the best model (the line of best fit), and basically if the best
model is any good then it should fit the data significantly better than our basic model:
deviation =
X
ðobserved − modelÞ2
(7.3)
This is all quite abstract so let’s look at an example. Imagine that I was
interested in predicting record sales (Y) from the amount of money spent
advertising that record (X). One day my boss came into my office and said
‘Andy, I know you wanted to be a rock star and you’ve ended up working
as my stats-monkey, but how many records will we sell if we spend
£100,000 on advertising?’ If I didn’t have an accurate model of the relationship
between record sales and advertising, what would my best guess
be? Well, probably the best answer I could give would be the mean number
of record sales (say, 200,000) because on average that’s how many records
we expect to sell. This response might well satisfy a brainless record company
executive (who didn’t offer my band a record contract). However,
what if he had asked ‘How many records will we sell if we spend £1 on advertising?’ Again, in
the absence of any accurate information, my best guess would be to give the average number
of sales (200,000). There is a problem: whatever amount of money is spent on advertising I
How do I tell if my
model is good?
176 DISCOVERING STATISTICS USING SAS
always predict the same level of sales. As such, the mean is a model of ‘no relationship’ at all
between the variables. It should be pretty clear then that the mean is fairly useless as a model
of a relationship between two variables – but it is the simplest model available.
So, as a basic strategy for predicting the outcome, we might choose to use the mean, because
on average it will be a fairly good guess of an outcome, but that’s all. Using the mean as a model,
we can calculate the difference between the observed values, and the values predicted by the
mean (equation (7.3)). We saw in section 2.4.1 that we square all of these differences to give
us the sum of squared differences. This sum of squared differences is known as the total sum of
squares (denoted SST
) because it is the total amount of differences present when the most basic
model is applied to the data. This value represents how good the mean is as a model of the
observed data. Now, if we fit the more sophisticated model to the data, such as a line of best fit,
we can again work out the differences between this new model and the observed data (again
using equation (7.3)). In the previous section we saw that the method of least squares finds
the best possible line to describe a set of data by minimizing the difference between the model
fitted to the data and the data themselves. However, even with this optimal model there is still
some inaccuracy, which is represented by the differences between each observed data point and
the value predicted by the regression line. As before, these differences are squared before they
are added up so that the directions of the differences do not cancel out. The result is known as
the sum of squared residuals or residual sum of squares (SSR
). This value represents the degree
of inaccuracy when the best model is fitted to the data. We can use these two values to calculate
how much better the regression line (the line of best fit) is than just using the mean as a model
(i.e. how much better is the best possible model than the worst model?). The improvement
in prediction resulting from using the regression model rather than the mean is calculated by
calculating the difference between SST
and SSR
. This difference shows us the reduction in the
inaccuracy of the model resulting from fitting the regression model to the data. This improvement
is the model sum of squares (SSM
). Figure 7.4 shows each sum of squares graphically.
If the value of SSM
is large then the regression model is very different from using the
mean to predict the outcome variable. This implies that the regression model has made a big
improvement to how well the outcome variable can be predicted. However, if SSM
is small
then using the regression model is little better than using the mean (i.e. the regression model
is no better than taking our ‘best guess’). A useful measure arising from these sums of squares
is the proportion of improvement due to the model. This is easily calculated by dividing the
sum of squares for the model by the total sum of squares. The resulting value is called R2
and
to express this value as a percentage you should multiply it by 100. R2
represents the amount
of variance in the outcome explained by the model (SSM
) relative to how much variation
there was to explain in the first place (SST
). Therefore, as a percentage, it represents the percentage
of the variation in the outcome that can be explained by the model:
R2
=
SSM
SST
(7.4)
This R2
is the same as the one we met in Chapter 6 (section 6.5.2.3) and you might have
noticed that it is interpreted in the same way. Therefore, in simple regression we can take
the square root of this value to obtain Pearson’s correlation coefficient. As such, the correlation
coefficient provides us with a good estimate of the overall fit of the regression
model, and R2
provides us with a good gauge of the substantive size of the relationship.
A second use of the sums of squares in assessing the model is through the F-test.
I mentioned way back in Chapter 2 that test statistics (like F) are usually the amount of
systematic variance divided by the amount of unsystematic variance, or, put another way,
the model compared against the error in the model. This is true here: F is based upon the
ratio of the improvement due to the model (SSM
) and the difference between the model
and the observed data (SSR
). Actually, because the sums of squares depend on the number
177CHAPTER 7 Regression
of differences that we have added up, we use the average sums of squares (referred to as
the mean squares or MS). To work out the mean sums of squares we divide by the degrees
of freedom (this is comparable to calculating the variance from the sums of squares – see
section 2.4.2). For SSM
the degrees of freedom are simply the number of variables in the
model, and for SSR
they are the number of observations minus the number of parameters
being estimated (i.e. the number of beta coefficients including the constant). The result is
the mean squares for the model (MSM
) and the residual mean squares (MSR
). At this stage
it isn’t essential that you understand how the mean squares are derived (it is explained in
Chapter 10). However, it is important that you understand that the F-ratio (equation (7.5))
is a measure of how much the model has improved the prediction of the outcome compared
to the level of inaccuracy of the model:
F =
MSM
MSR
(7.5)
Figure 7.4
Diagram showing
from where the
regression sums
of squares derive
178 DISCOVERING STATISTICS USING SAS
If a model is good, then we expect the improvement in prediction due to the model to be
large (so MSM
will be large) and the difference between the model and the observed data to
be small (so MSR
will be small). In short, a good model should have a large F-ratio (greater
than 1 at least) because the top of equation (7.5) will be bigger than the bottom. The exact
magnitude of this F-ratio can be assessed using critical values for the corresponding degrees
of freedom (as in the Appendix).
7.2.4.   Assessing individual predictors 1
We’ve seen that the predictor in a regression model has a coefficient (b1
), which in simple
regression represents the gradient of the regression line. The value of b represents
the change in the outcome resulting from a unit change in the predictor. If the model
was useless at predicting the outcome, then if the value of the predictor changes, what
might we expect the change in the outcome to be? Well, if the model is very bad then
we would expect the change in the outcome to be zero. Think back to Figure 7.4 (see
the panel representing SST
) in which we saw that using the mean was a very bad way of
predicting the outcome. In fact, the line representing the mean is flat, which means that
as the predictor variable changes, the value of the outcome does not change (because
for each level of the predictor variable, we predict that the outcome will equal the
mean value). The important point here is that a bad model (such as the mean) will have
regression coefficients of 0 for the predictors. A regression coefficient of 0 means: (1)
a unit change in the predictor variable results in no change in the predicted value of
the outcome (the predicted value of the outcome does not change at all); and (2) the
gradient of the regression line is 0, meaning that the regression line is flat. Hopefully,
you’ll see that it logically follows that if a variable significantly predicts an outcome,
then it should have a b-value significantly different from zero. This hypothesis is tested
using a t-test (see Chapter 9). The t-statistic tests the null hypothesis that the value of b
is 0: therefore, if it is significant we gain confidence in the hypothesis that the b-value
is significantly different from 0 and that the predictor variable contributes significantly
to our ability to estimate values of the outcome.
Like F, the t-statistic is also based on the ratio of explained variance to unexplained
variance or error. Well, actually, what we’re interested in here is not so much variance but
whether the b we have is big compared to the amount of error in that estimate. To estimate
how much error we could expect to find in b we use the standard error. The standard error
tells us something about how different b-values would be across different samples. We
could take lots and lots of samples of data regarding record sales and advertising budgets
and calculate the b-values for each sample. We could plot a frequency distribution of these
samples to discover whether the b-values from all samples would be relatively similar, or
whether they would be very different (think back to section 2.5.1). We can use the standard
deviation of this distribution (known as the standard error) as a measure of the similarity
of b-values across samples. If the standard error is very small, then it means that most
samples are likely to have a b-value similar to the one in our sample (because there is little
variation across samples). The t-test tells us whether the b-value is different from 0 relative
to the variation in b-values across samples. When the standard error is small even a small
deviation from zero can reflect a meaningful difference because b is representative of the
majority of possible samples.
Equation (7.6) shows how the t-test is calculated and you’ll find a general version of
this equation in Chapter 9 (equation (9.1)). The bexpected
is simply the value of b that we
would expect to obtain if the null hypothesis were true. I mentioned earlier that the null
179CHAPTER 7 Regression
hypothesis is that b is 0 and so this value can be replaced by 0. The equation simplifies to
become the observed value of b divided by the standard error with which it is associated:
t =
bobserved − bexpected
SEb
=
bobserved
SEb
(7.6)
The values of t have a special distribution that differs according to the degrees of freedom
for the test. In regression, the degrees of freedom are N − p − 1, where N is the total sample
size and p is the number of predictors. In simple regression when we have only one predictor,
this reduces down to N − 2. Having established which t-distribution needs to be used,
the observed value of t can then be compared to the values that we would expect to find if
there was no effect (i.e. b = 0): if t is very large then it is unlikely to have occurred when
there is no effect (these values can be found in the Appendix). SAS provides the exact probability
that the observed value (or a larger one) of t would occur if the value of b was, in
fact, 0. As a general rule, if this observed significance is less than .05, then scientists assume
that b is significantly different from 0; put another way, the predictor makes a significant
contribution to predicting the outcome.
7.3.  Doing simple regression on SAS 1
So far, we have seen a little of the theory behind regression, albeit restricted to the situation
in which there is only one predictor. To help clarify what we have learnt so far,
we will go through an example of a simple regression on SAS. Earlier on I asked you
to imagine that I worked for a record company and that my boss was interested in predicting
record sales from advertising. There are some data for this example in the file
Record1.sas7bdat. This data file has 200 rows, each one representing a different record.
There are also two columns, one representing the sales of each record in the week after
release and the other representing the amount (in pounds) spent promoting the record
before release. This is the format for entering regression data: the outcome variable and
any predictors should be entered in different columns, and each row should represent
independent values of those variables.
The pattern of the data is shown in Figure 7.5 and it should be clear that a positive
relationship exists: so, the more money spent advertising the record, the more it is likely
to sell. Of course there are some records that sell well regardless of advertising (top left
of scatterplot), but there are none that sell badly when advertising levels are high (bottom
right of scatterplot). The scatterplot also shows the line of best fit for these data: bearing in
mind that the mean would be represented by a flat line at around the 200,000 sales mark,
the regression line is noticeably different.
To find out the parameters that describe the regression line, and to see whether this line
is a useful model, we need to run a regression analysis. To do the analysis you need to use
PROC REG.
Proc reg syntax is very straightforward, and is shown in SAS Syntax 7.1. The model
statement is written in the form of the equation: we think that sales are a function of adverts,
so we write sales = adverts. Notice that Proc Reg is slightly different to other procedures,
because we need to write quit; after the run statement.
180 DISCOVERING STATISTICS USING SAS
AdvertsingBudget(thousandsofpounds)
0100020003000
Record Sales (thousands)
0 100 200 300 400
Figure 7.5
Scatterplot
showing the
relationship
between record
sales and the
amount spent
promoting the
record
PROC REG DATA=chapter6.record1;
	 MODEL sales = adverts;
	 RUN;
QUIT;
SAS Syntax 7.1
7.4.  Interpreting a simple regression 1
7.4.1.   Overall fit of the model 1
The output from the regression is shown in SAS Output 7.1. The first part of the output
reports an analysis of variance (ANOVA – see Chapter 10). The summary table shows the
various sums of squares described in Figure 7.4 and the degrees of freedom associated with
each. From these two values, the average sums of squares (the mean squares) can be calculated
by dividing the sums of squares by the associated degrees of freedom. The most important
part of the table is the F-ratio, which is calculated using equation (7.5), and the associated
significance value of that F-ratio. For these data, F is 99.59, which is significant at p < .001
(because the value in the column labelled Sig. is less than .001). Researchers usually dont
report p-values below 0.001 though. This result tells us that there is less than a 0.1% chance
that an F-ratio at least this large would happen if the null hypothesis were true. Therefore,
we can conclude that our regression model results in significantly better prediction of record
sales than if we used the mean value of record sales. In short, the regression model overall
predicts record sales significantly well.
The second part of the output is a summary of the model. This summary table provides
the value of R and R2
for the model that has been derived (as well as some other things we
are not going to worry about for now). For these data, R has a value of .578 and because
there is only one predictor, this value represents the simple correlation between advertising
181CHAPTER 7 Regression
and record sales (you can confirm this by running a correlation using what you were taught
in Chapter 6). The value of R2
is .335, which tells us that advertising expenditure can
account for 33.5% of the variation in record sales. In other words, if we are trying to
explain why some records sell more than others, we can look at the variation in sales of different
records. There might be many factors that can explain this variation, but our model,
which includes only advertising expenditure, can explain approximately 33% of it. This
means that 67% of the variation in record sales cannot be explained by advertising alone.
Therefore, there must be other variables that have an influence also.
7.4.2.   Model parameters 1
The ANOVA tells us whether the model, overall, results in a significantly
good degree of prediction of the outcome variable. However, the ANOVA
doesn’t tell us about the individual contribution of variables in the model
(although in this simple case there is only one variable in the model and
so we can infer that this variable is a good predictor. The third part of
the output provides details of the model parameter estimates (the beta
values) and the significance of these values. We saw in equation (7.2)
that b0
was the Y intercept and this value is the value labelled Parameter
Estimate (in the SAS output) for the constant. So, from the table, we
SAS Output 7.1
How do I interpret
b values?
The REG Procedure
Model: MODEL1
Dependent Variable: SALES Record Sales (thousands)
Number of Observations Read 200
Number of Observations Used 200
Analysis of Variance
Source DF
Sum of
Squares
Mean
Square F Value Pr > F
Model 1 433688 433688 99.59 <.0001
Error 198 862264 4354.86953
Corrected Total 199 1295952
Root MSE 65.99144 R-Square 0.3346
Dependent Mean 193.20000 Adj R-Sq 0.3313
Coeff Var 34.15706
Parameter Estimates
Variable Label DF
Parameter
Estimate
Standard
Error t Value Pr > |t|
Intercept Intercept 1 134.13994 7.53657 17.80 <.0001
ADVERTS Advertsing Budget (thousands of
pounds)
1 0.09612 0.00963 9.98 <.0001
182 DISCOVERING STATISTICS USING SAS
can say that b0
is 134.14, and this can be interpreted as meaning that when no money is
spent on advertising (when X = 0), the model predicts that 134,140 records will be sold
(remember that our unit of measurement was thousands of records). We can also read off
the value of b1
from the table and this value represents the gradient of the regression line.
It is 0.096. Although this value is the slope of the regression line, it is more useful to think
of this value as representing the change in the outcome associated with a unit change in the
predictor. Therefore, if our predictor variable is increased by one unit (if the advertising
budget is increased by 1), then our model predicts that 0.096 extra records will be sold.
Our units of measurement were thousands of pounds and thousands of records sold, so
we can say that for an increase in advertising of £1000 the model predicts 96 (0.096 ×
1000 = 96) extra record sales. As you might imagine, this investment is pretty bad for the
record company: it invests £1000 and gets only 96 extra sales! Fortunately, as we already
know, advertising accounts for only one-third of record sales.
We saw earlier that, in general, values of the regression coefficient b represent the change
in the outcome resulting from a unit change in the predictor and that if a predictor is having
a significant impact on our ability to predict the outcome then this b should be different
from 0 (and big relative to its standard error). We also saw that the t-test tells us whether
the b-value is different from 0. SAS provides the exact probability that the observed value of
t would occur if the value of b in the population were 0. If this observed significance is less
than .05, then scientists agree that the result reflects a genuine effect (see Chapter 2). For
these two values, the probabilities are <.0001 and so we can say that the probability of these
t values (or larger) occurring if the values of b in the population were 0 is less than .0001.
Therefore, the bs are different from 0 and we can conclude that the advertising budget
makes a significant contribution (p < .0001) to predicting record sales.
SELF-TEST How is the t in SAS Output 7.1 calculated?
Use the values in the table to see if you can get the
same value as SAS.
7.4.3.   Using the model 1
So far, we have discovered that we have a useful model, one that significantly improves
our ability to predict record sales. However, the next stage is often to use that model to
make some predictions. The first stage is to define the model by replacing the b-values in
equation (7.2) with the values from SAS Output 7.1. In addition, we can replace the X and
Y with the variable names so that the model becomes:
record salesi = b0 + b1advertising budgeti
= 134:14 + ð0:096 × advertising budgetiÞ
(7.7)
It is now possible to make a prediction about record sales, by replacing the advertising
budget with a value of interest. For example, imagine a record executive wanted to spend
£100,000 on advertising a new record. Remembering that our units are already in thousands
of pounds; we can simply replace the advertising budget with 100. He would discover
that record sales should be around 144,000 for the first week of sales: