Last week

1
Last week

• How to run chi-square tests of association on
  SPSS.
• When the data are scarce, the usual chi-square
  test can give a misleading result.
• If there are warnings about low expected
  frequencies, run an EXACT TEST.

2
Regression

• Regression is the other side of the associative
  coin.
• We use a correlation coefficient to measure the
  STRENGTH of an association.
• We can use regression to PREDICT values of one
  variable from those of the other.

3
The regression line of Violence upon Preference

• The REGRESSION LINE is the line that fits the
  points best according to the LEAST SQUARES
  criterion.

4
The regression line
Y (Violence)
P
Q
2.091
intercept
(0, 0)
X (Exposure)
5
Errors or residuals
Y (Violence)
Intercept 2.091
(0, 0)
9
X (Exposure)
6
Variance accounted for

• An important aspect of regression is accounting
  for VARIANCE in the target or criterion variable
  in terms of regression upon the independent
  variable or predictor.

7
The total variance of the target variable
MY
These are the deviations from the MY line from
which the total variance of the scores on the
target variable can be calculated.
8
Prediction without regression

• When the variables show no association, the slope
  of the regression line is zero and the line runs
  horizontally through the mean MY of the criterion
  or dependent variable.
• The intercept (B0) is MY in this case.

9
Deviations from MY of points on the regression
line
From these deviations, the variance attributable
to regression can be calculated.
10
Regression line residuals
Here are the errors or residuals when regression
is used. A variance estimate can be calculated
from the residuals.
11
Variance accounted for by regression

• The deviations from MY of points on the line are
  the basis of the variance of the criterion
  variable Y accounted for by regression on X.

12
Coefficient of determination

• The PROPORTION of the variance of the target or
  criterion variable (Actual Violence) accounted
  for by regression on the regressor or independent
  variable (Exposure) is known as the COEFFICIENT
  OF DETERMINATION and is the square of the Pearson
  correlation.

13
Coefficient of determination (r2)
14
In our example
15
Theoretical importance of r2

• Often the predictions of individual cases from
  regression arent very accurate.
• Researchers are usually more interested in the
  coefficient of determination, because it provides
  a measure of the extent to which a predictor or
  regressor variable accounts for variance in the
  target variable.
• If r2 is high, the inference is that the
  abilities involved in the regressor are
  implicated in the target variable.

16
Lecture 11RUNNING SIMPLE REGRESSION ON
SPSS
17
Finding regression
18
Saving the predictions and residuals

• Its useful to save the predicted values and
  residuals (errors) from the regression equation.
• Click the Save button at the foot of the Linear
  Regression dialog.

Click the Savebutton to save the predicted
values.
19
The regression coefficients

• The only entries that concern us just now are
  those under the B column (Unstandardised
  Coefficients).
• We see that B0 2.091 and B 0.736. These are,
  respectively, the intercept and the slope of the
  regression equation.

20
Coefficient of determination

• The values given for the Pearson correlation and
  the coefficient of determination r2 have been
  ringed.

21
Data View

• I have renamed the estimates of Y and the
  residuals in the two rightmost columns.
• Notice that the residuals are Actual Violence
  scores minus the predictions from regression.
• When a residual has a negative sign, it means
  that the actual Y value was below the regression
  line.

22
Importance of regression

• Regression is used extensively in some areas of
  psychological research, such as health
  psychology.
• You will be hearing much more about it next year.
  

23
Revision of the normal curve
24
Hypothesis testing

• In my final lecture, I shall be considering the
  t-testing procedure that you used to analyse the
  data from the inverted faces and landscapes
  experiment.
• This will involve a more general discussion of
  HYPOTHESIS TESTING.
• This afternoon, I shall try to lay the
  foundations.

25
Normal distribution

• A NORMAL DISTRIBUTION is symmetrical and
  bell-shaped.
• If a variable is normally distributed, 95 of
  values lie within 1.96 standard deviations (2
  approx.) on EITHER side of the mean.

0.95 (95)
2 ½ .025
2 ½ .025
mean
mean 1.96SD
mean 1.96SD
26
Specifying a normal distribution

• Suppose that a variable X has a normal
  distribution with mean µ and standard deviation
  s.
• We write this as shown.

27
The standard normal variable z

• If X is a normal variable, that is,
• XN(µ,s),
• z will also be normally distributed.
• z is known as the STANDARD NORMAL VARIABLE.

28
Standardisation

• Strictly speaking, z is defined in relation to
  the theoretical normal population mean µ.
• However, any set of scores X can be STANDARDISED
  by subtracting the sample mean from each score
  and dividing by the sample standard deviation.

29
Effects of standardisation

• Standardising a set of scores (or a population
  of scores) has two effects
• The mean becomes zero
• The standard deviation becomes 1.

30
The standard normal distribution

• In the notation I introduced earlier, we can
  represent the standard normal distribution as
  follows.

31
Distribution of z

• Standardising a set of scores does NOT make them
  normally distributed.
• If theres a tail to the right (ve skew) before
  transforming X to z, there will be one after the
  transformation.
• Nevertheless, whatever the shape of the original
  distribution, the mean standardised score will be
  zero and the standard deviation will be 1.

32
Referring to z

• A question about the probability of a range of
  values of ANY normally distributed variable X
  with mean µ and standard deviation s can be
  translated into a question about an equivalent
  range of the standard normal variable z.

33
Using z
Probability that X (IQ) lies between 70 and
130 AND ALSO Probability that z lies between
-1.96 and 1.96.
Probability that X (IQ) is at least 130 AND ALSO
Probability that z is at least 1.96
0.95
X (IQ) 70 100
130 100 1.96SD z
-1.96 0
1.96
34
Referring questions from X to z

• What is the probability of an IQ of at least 130?
• This is to ask about the probability that X is at
  least 130, where X N(100, 15).
• Transform X to z z (130 100)/15 2.
• We know from memory that the probability of z
  greater than 2 (actually 1.96) .025.

35
Probability of an IQ between 100 and 130?

• Convert these values to values of z.
• If X 100, z 0.
• If X 30, z 2.
• Pr(z between 0 and 2) 0.95/2 0.475.

0.95 (95)
2 ½ .025
2 ½ .025
µ
µ 1.96SD
X µ 1.96SD
z -1.96 0 1.96
36
Finding the probability of a range of values of
X

• In the problems we have considered, the value of
  z has always been around 2 (about 1.96), so that
  we can find the probability from memory.
• Suppose z 1, 0.5, or any value other than
  1.96?
• Just standardise the value of X by converting it
  to z z (X mean)/SD.
• The are available tables in standard statistics
  textbooks which give probabilities of ANY
  specified range of values of z. You can also use
  the SPSS cumulative distribution function CDF to
  find such probabilities.

37
Tables

• There are countless normal distributions.
• But there is only ONE standard normal
  distribution, to which any of the others can be
  transformed by z (X mean)/SD.
• So only the probabilities of ranges of values of
  z need to be tabled.
• It would not be feasible to table the
  probabilities for ALL possible normal
  distributions.

38
To sum up

• If we know the DISTRIBUTION of some variable, we
  can assign a probability of obtaining a value
  within a specified range.
• We can visualise the probability of such a value
  as the area under the curve of the distribution.
• If the distribution is normal, we can translate
  probability questions in the original units of
  measurement into questions about ranges of z,
  which, provided X is normally distributed, has
  the STANDARD NORMAL DISTRIBUTION.

39
Random variables

• A RANDOM VARIABLE is defined in the context of an
  experiment of chance, such as rolling a die or
  tossing a coin.
• Let X be the result of rolling a die. X is a
  random variable.
• Let Y be the number of heads in 10 tosses of a
  coin. Y is a random variable.

40
Gathering data

• The gathering of data is an experiment of chance.
  
• Essentially, you are SELECTING observations at
  random from POPULATIONS of possible observations.
• The value of a variable that you are observing is
  a RANDOM variable.
• In the Caffeine experiment, the value of a score
  under the Caffeine condition is a random
  variable, as is that of a score under the Placebo
  condition.

41
Drawing tickets

• This barrel contains numbered tickets.
• I draw a ticket at random. Let X be the number
  on the ticket. X is a random variable.

42
Drawing tickets

• Early studies of probability involved drawing
  real tickets from real barrels. Nowadays, we use
  the computer.
• Computers enable us to specify the distribution
  from which we are drawing our ticket.

43
Using the computer

• The 4000 IQs were sampled by first placing 4000
  numbers in the column named Ones. Any numbers
  will do.
• Now we are going to draw another sample of 4000
  IQs from the same population.

44
Resampling from the IQ population
45
Descriptives

• I put some more ones in the first column 48,
  000, actually!
• As expected with samples this size, their means
  and SDs are very similar and close to the
  theoretical mean of 100 and standard deviation of
  15.

46
Independent random variables

• Here are two barrels, each containing numbered
  tickets.
• Let X be the number of the ticket that I draw
  from the first barrel. Let Y be the number of the
  ticket that I draw from the second barrel.
• X and Y are INDEPENDENT random variables, because
  the value of X does not determine the value of Y
  or vice versa.

Let X be a value drawn from this barrel.
Let Y be a value drawn from this barrel.
47
The sum of random variables

• X and Y are random variables.
• Let SUM X Y
• SUM is also a random variable.

X
Y
SUM X Y
48
The variance of the sum

• The variance of the sum of INDEPENDENT random
  variables is the sum of their variances.

X Y
Y
X
49
Create the SUM

• In a fourth column, the sum of IQ and IQtwo will
  appear.
• You can see that the variance of SUM is very
  close to twice the variance of either IQ or
  IQtwo.

50
Taking a sample
51
Variance of S X
52
Adding and multiplying by a constant
s2
Adding a constant k
M
M k
k2s2
s2
Multiplying by a constant k
M
kM
53
Effect upon the variance of multiplying by a
constant

• When you multiply by a constant, you multiply the
  variance by the SQUARE of the constant.

54
Variance of the mean
The variance of the means is the original
variance s2 divided by n, the size of the
sample.
55
The variance of the mean

• The barrel on the right contains means of samples
  of size n drawn from the barrel on the left.
• The variance of the means is the original
  variance divided by n.

X
M
s2
s2/n
56
Standard error of the mean

• The STANDARD ERROR OF THE MEAN sM is the square
  root of the variance of the mean.

57
Sampling distribution

• The distribution of a STATISTIC such as the mean
  is known as its SAMPLING DISTRIBUTION.
• If X is normally distributed, then the mean M is
  also normally distributed.
• The sampling distribution of M is normal and
  centred on the mean µ of the variable X.
• The variance of the sampling distribution of M is
  s2/n.
• The standard deviation of the sampling
  distribution of the mean (the standard error of
  the mean) sM is s/vn .

58
Sampling distribution of the mean
59
Effect of increasing n

• IQ N(100, 15)
• Let M4, M16 and M64 be means of samples of size n
  4, 16 and 64 respectively.
• The values of sM are, respectively, 7.5, 3.75 and
  1.88.

60
Effect of sample size n
61
Effect of increasing the sample size n
µ
62
Using z
Probability that X (IQ) lies between 70 and
130 AND ALSO Probability that z lies between
-1.96 and 1.96.
Probability that X (IQ) is at least 130 AND ALSO
Probability that z is at least 1.96
0.95
X s 1.96s µ
µ 1.96s M µ
1.96sM µ µ 1.96sM z
-1.96 0
1.96
63
Referring to z

• A question about a range of values of ANY
  normally distributed variable can always be
  translated into a question about a range of
  values of the standard normal variable z.
• Just subtract the mean and divide by the standard
  deviation.
• If your question is about a range of values for
  the MEAN, you must divide by the STANDARD ERROR,
  not the original population SD.

64
Question

• If I select 9 IQs at random and take their mean
  M, what is the probability that M is at least
  110?

65
Answer
66
Important

• If your question is about MEANS, divide by the
  STANDARD ERROR OF THE MEAN sM, not the standard
  deviation of the original population.

67
The difference between 2 random variables

• Let D X Y
• D is also a random variable.

X
Y
D X Y
68
The variance of the difference

• The variance of the difference between 2
  independent random variables is the SUM of their
  variances.

D X -Y
Y
X
69
Explanation
70
Demonstration

• The Descriptives procedure shows that the
  variance of the difference is approximately equal
  to the variance of the sum.
• In the population, they are EXACTLY equal.

71
Summary

• Review of regression.
• The COEFFICIENT OF DETERMINATION r2 is the
  proportion of the variance of the target,
  criterion or dependent variable accounted for by
  regression.
• The gathering of data is an EXPERIMENT OF CHANCE,
  to which the concept of RANDOM VARIABLE is
  applicable.

72
Summary

• The variance of the sum of INDEPENDENT random
  variables (separate barrels) is the sum of their
  variainces.
• The variance of the difference between 2 random
  variables is also the SUM of their variances.
• The distribution of means is the SAMPLING
  DISTRIBUTION OF THE MEAN.
• The STANDARD ERROR OF THE MEAN is the SD of the
  sampling distribution of the mean.

73
Summary

• If the parent population is normal, so is the
  sampling distribution of the mean.
• A question about the probability of a range of
  values of ANY normal variable can be referred to
  a question about ranges of z, the STANDARD NORMAL
  VARIABLE.
• When standardising a value of M, however, use the
  standard ERROR, not the SD of the parent
  population.

74
Question

• What is the probability that a random sample of
  16 IQs will have a mean between 92.5 and 100?