Assumptions Underlying Parametric Statistical Techniques

1
Chapter 13

• Assumptions Underlying Parametric Statistical
  Techniques

2
Parametric Statistics

• We have been studying parametric statistics.
• They include estimations of mu and sigma,
  correlation, t tests and F tests.

3
Five Assumptions
To validly use parametric statistics, we make

• two research assumptions
• two assumptions about the type of the
  distributions in the samples,
• and one assumption about the kind of numbering
  system that we are using.

4
Research Assumptions

• Subjects have to be randomly selected from the
  population.
• Experimental error is randomly distributed across
  samples in the design.
• (We will not discuss these any further).

5
Distribution Assumptions

• The distribution of sample means fit a normal
  curve.
• Homogeneity of variance (using FMAX).

6
Assumptions about Numbering Schemes

• The measures we take are on an interval scale.
• (Other numbering scales, such as ordinal and
  nominal, do not allow the estimation of
  population parameters such as mu and sigma and
  the tests used to analyze such data are therefore
  call nonparametric).

7
Violating the Assumptions
If any of these assumptions are violated, we
cannot use parametric statistics. We must use
less-powerful, non-parametric statistics.
8
Sample Means Form a Normal Curve
9
Sample Means

• An assumption we need to make is that the
  distribution of sample means is normally
  distributed.
• This is not as extreme an assumption as it might
  seem.
• We will follow the example in the book to
  demonstrate (only smaller).

10
An Artificial Population
Subject Score

• Seven subjects.
• Each subject has a different score.
• We sample five subjects.

A 1 B 2 C 3 D 4 E 5 F 6 G 7
11
The Distribution is Rectangular
FREQUENCY
3 2 1 0
? ? ? ? ? ? ?
1 2 3 4 5 6 7
SCORE
12
All Possible Samples
Sample Scores Mean
Sample Scores Mean
ABCDE 12345 3.0 ABCDF 12346 3.2 ABCDG 12347
3.4 ABCEF 12356 3.4 ABCEG 12357 3.6 ABCFG
12367 3.8 ABDEF 12456 3.6 ABDEG 12457
3.8 ABDFG 12467 4.0 ABEFG 12567 4.2
ACDEF 13456 3.8 ACDEG 12457 4.0 ACDFG 13467
4.2 ACEFG 13567 4.4 ADEFG 14567 4.6 BCDEF
23456 4.0 BCDEG 23457 4.2 BCDFG 23467
4.4 BCEFG 23567 4.6 BDEFG 24567 4.8 CDEFG
34567 5.0
13
Sample Distribution
14
Normal Curve for Sample Means Conclusion

• Even if we have a small population (7),
• with a rectangular distribution,
• and a small sample size (5),
• which yields a small number of possible samples
  (21),
• the sample means tend to fall in an
  (approximately) normal distribution.
• This assumption that the distribution of sample
  means will basically fit a normal curve is seldom
  violated.
• This assumption is robust.

15
But it can happen -Violating the Normal Curve
Assumption
Distributions of sample means can vary from
normal in several ways.

• Normal curves
• are symmetric
• are bell-shaped
• have a single peak

• Non-normal curves
• have skew
• have kurtosis- platykutic or leptokurtic
• are polymodal

16
Symmetry
F r e q u e n c y
score
17
Skewed
NORMAL
18
Bell-shaped
F r e q u e n c y
1 SD is 34 2 SD is 48 etc.
score
19
Kurtosis
NORMAL
20
One mode
F r e q u e n c y
score
There is only one mode and it equals the
median and the mean.
21
Polymodality
NORMAL
22
Violation of normally distributed sample means

• If the distribution of sample means is
• skewed,
• or has kurtosis,
• or more than one mode,
• then we cannot use parametric statistics.
• BUT THIS IS RARE.

23
Homogeneity of Variance and FMAX
24
For F Ratios and t Tests

• We assume that the distribution of scores around
  each sample mean is similar.
• The distributions within each group all estimate
  the same thing, that is, sigma2.
• The mean squares within each group should be the
  approximately the same in each group, differing
  only because of random sampling fluctuation.
• For F ratios and t tests, this is called
  homogeneity of variance.

25
For Correlation

• For correlation, the scores must vary roughly the
  same amount around the entire length of the
  regression line.
• This is called homoscedasticity.

26
Homoscedasticity
27
Non-Homoscedasticity
28
Homogeneity of Variance

• In mathematical terms, homogeneity of variance
  means that the mean squares for each group are
  the same.

We use the FMAX test to check if the group with
the smallest mean squares is too different from
the group with the largest mean squares.
29
FMAX

• If FMAX is significant, then the Mean Squares
  deviate from each other too much.
• The assumption of homogeneity of variance is
  violated.
• We cannot use parametric statistics!

30
Why???

• Because all parametric statistical procedures
  rely on our ability to estimate sigma2 with MSW.
• If the estimates of MSW among the grous differ
  among groups so that Fmax is significant, the
  odds are someone (most likely the senior
  experimenter) messed up and created a measure
  with too small a range of scores.

31
When that happens all the scores pile up at one
end of the scale.

• When everyone scores at the top or bottom a
  scale, individual differences and measurement
  problems seem to disappear.
• We call this a ceiling effect (if the scores are
  all at the top of the scale) and a floor effect
  if the scores are all at the bottom

32

• Because ID and MP in one or more groups have been
  pushed up against the top or bottom of the scale
  there is practically no within group variation.
• So, while adding df, the group contributes little
  or nothing to sum of squares within group (SSW).
• So, when you include one or more groups with
  practically no variation within group in your
  totals sums of squares and mean square, you wind
  up with an underestimate of sigma2.
• This makes it possible to get significant results
  not because you have pushed the means apart with
  an IV, but because MSW is an underestimat

33

• This makes it possible to get significant results
  not because you have pushed the means apart with
  an IV, but because MSW is an underestimate of
  sigma2 and therefore the denominator of the F or
  t test will be too small.
• So you can get significant results more often
  than you should when the null is true.

34
Uncrowded vs crowded groups How crowded do you
feel?
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
1 2 3 5 5 6 6 4
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8
9 8 9 9 9 9 8 9
35
FMAX

• In FMAX, the MAX part refers to the largest
  ratio that can be obtained by comparing the
  estimated variances from 2 experimental groups.

The significance of FMAX is checked in an FMAX
table.
36
K number of variances
2 3 4 5 6
7 8 9 10 4 23.2 37
49 59 69 79 89 97
106 5 14.9 22 28 33 38
42 46 50 54 6 11.1 15.5
19.1 22 25 27 30 32
34 7 8.89 12.1 14.5 16.5 18.4 20
22 23 24 8 7.50 9.9
11.7 13.2 14.5 15.8 16.9 17.9 18.9 9
6.54 8.5 9.9 11.1 12.1 13.1
13.9 14.7 15.3 10 5.85 7.4 8.6
9.6 10.4 11.1 11.8 12.4 12.9 12
4.91 6.1 6.9 7.6 8.2 8.7
9.1 9.5 9.9 15 4.07 4.9 5.5
6.0 6.4 6.7 7.1 7.3
7.5 20 3.32 3.8 4.3 4.6 4.9
5.1 5.3 5.5 5.6 30 2.63
3.0 3.3 3.4 3.6 3.7 3.8
3.9 4.0 60 1.96 2.2 2.3 2.4
2.4 2.5 2.5 2.6 2.6
dfFMAX
alpha .01.
37
The critical values.
k number of variances
2 3 4 5 6
7 8 9 10 4 23.2 37
49 59 69 79 89 97
106 5 14.9 22 28 33 38
42 46 50 54 6 11.1 15.5
19.1 22 25 27 30 32
34 7 8.89 12.1 14.5 16.5 18.4 20
22 23 24 8 7.50 9.9
11.7 13.2 14.5 15.8 16.9 17.9 18.9 9
6.54 8.5 9.9 11.1 12.1 13.1
13.9 14.7 15.3 10 5.85 7.4 8.6
9.6 10.4 11.1 11.8 12.4 12.9 12
4.91 6.1 6.9 7.6 8.2 8.7
9.1 9.5 9.9 15 4.07 4.9 5.5
6.0 6.4 6.7 7.1 7.3
7.5 20 3.32 3.8 4.3 4.6 4.9
5.1 5.3 5.5 5.6 30 2.63
3.0 3.3 3.4 3.6 3.7 3.8
3.9 4.0 60 1.96 2.2 2.3 2.4
2.4 2.5 2.5 2.6 2.6
dfFMAX
38
Book Example
39
k number of variances
2 3 4 5 6
7 8 9 10 4 23.2 37
49 59 69 79 89 97
106 5 14.9 22 28 33 38
42 46 50 54 6 11.1 15.5
19.1 22 25 27 30 32
34 7 8.89 12.1 14.5 16.5 18.4 20
22 23 24 8 7.50 9.9
11.7 13.2 14.5 15.8 16.9 17.9 18.9 9
6.54 8.5 9.9 11.1 12.1 13.1
13.9 14.7 15.3 10 5.85 7.4 8.6
9.6 10.4 11.1 11.8 12.4 12.9 12
4.91 6.1 6.9 7.6 8.2 8.7
9.1 9.5 9.9 15 4.07 4.9 5.5
6.0 6.4 6.7 7.1 7.3
7.5 20 3.32 3.8 4.3 4.6 4.9
5.1 5.3 5.5 5.6 30 2.63
3.0 3.3 3.4 3.6 3.7 3.8
3.9 4.0 60 1.96 2.2 2.3 2.4
2.4 2.5 2.5 2.6 2.6
dfFMAX
FMAX 16.33 gt 8.89 FMAX exceeds the critical
value. We cannot use parametric statistics.
40
Examples
Number Subjects Critical value Design of Means
in larger NG of FMAX 2X4 8
21 5.3 2X2
? 16
? 3X3 ?
11 ? 2X3
? 9
?
4
9
6
41
K number of variances
2 3 4 5 6
7 8 9 10 4 23.2 37
49 59 69 79 89 97
106 5 14.9 22 28 33 38
42 46 50 54 6 11.1 15.5
19.1 22 25 27 30 32
34 7 8.89 12.1 14.5 16.5 18.4 20
22 23 24 8 7.50 9.9
11.7 13.2 14.5 15.8 16.9 17.9 18.9 9
6.54 8.5 9.9 11.1 12.1 13.1
13.9 14.7 15.3 10 5.85 7.4 8.6
9.6 10.4 11.1 11.8 12.4 12.9 12
4.91 6.1 6.9 7.6 8.2 8.7
9.1 9.5 9.9 15 4.07 4.9 5.5
6.0 6.4 6.7 7.1 7.3
7.5 20 3.32 3.8 4.3 4.6 4.9
5.1 5.3 5.5 5.6 30 2.63
3.0 3.3 3.4 3.6 3.7 3.8
3.9 4.0 60 1.96 2.2 2.3 2.4
2.4 2.5 2.5 2.6 2.6
dfFMAX
42
Number Subjects Critical value Design of Means
in larger NG of FMAX 2X4 8
21 5.3 2X2
4 16
5.5 3X3 9
11 ? 2X3
6 9
?
43
K number of variances
2 3 4 5 6
7 8 9 10 4 23.2 37
49 59 69 79 89 97
106 5 14.9 22 28 33 38
42 46 50 54 6 11.1 15.5
19.1 22 25 27 30 32
34 7 8.89 12.1 14.5 16.5 18.4 20
22 23 24 8 7.50 9.9
11.7 13.2 14.5 15.8 16.9 17.9 18.9 9
6.54 8.5 9.9 11.1 12.1 13.1
13.9 14.7 15.3 10 5.85 7.4 8.6
9.6 10.4 11.1 11.8 12.4 12.9 12
4.91 6.1 6.9 7.6 8.2 8.7
9.1 9.5 9.9 15 4.07 4.9 5.5
6.0 6.4 6.7 7.1 7.3
7.5 20 3.32 3.8 4.3 4.6 4.9
5.1 5.3 5.5 5.6 30 2.63
3.0 3.3 3.4 3.6 3.7 3.8
3.9 4.0 60 1.96 2.2 2.3 2.4
2.4 2.5 2.5 2.6 2.6
dfFMAX
44
Number Subjects Critical value Design of Means
in larger NG of FMAX 2X4 8
21 5.3 2X2
4 16
5.5 3X3 9
11 12.4 2X3
6 9
?
45
K number of variances
2 3 4 5 6
7 8 9 10 4 23.2 37
49 59 69 79 89 97
106 5 14.9 22 28 33 38
42 46 50 54 6 11.1 15.5
19.1 22 25 27 30 32
34 7 8.89 12.1 14.5 16.5 18.4 20
22 23 24 8 7.50 9.9
11.7 13.2 14.5 15.8 16.9 17.9 18.9 9
6.54 8.5 9.9 11.1 12.1 13.1
13.9 14.7 15.3 10 5.85 7.4 8.6
9.6 10.4 11.1 11.8 12.4 12.9 12
4.91 6.1 6.9 7.6 8.2 8.7
9.1 9.5 9.9 15 4.07 4.9 5.5
6.0 6.4 6.7 7.1 7.3
7.5 20 3.32 3.8 4.3 4.6 4.9
5.1 5.3 5.5 5.6 30 2.63
3.0 3.3 3.4 3.6 3.7 3.8
3.9 4.0 60 1.96 2.2 2.3 2.4
2.4 2.5 2.5 2.6 2.6
dfFMAX
46
Number Subjects Critical value Design of Means
in larger NG of FMAX 2X4 8
21 5.3 2X2
4 16
5.5 3X3 9
11 9.5 2X3
6 9
14.5
47
Example other way
Number of Means 8 ? ? ?
MSG max 18.2 26.3 34.2 18.0
MSG min 1.1 2.0 4.6 0.5
FMAX 16.5 ? ? ?
Subjects in larger NG 10 12 21 7
dfFMAX 9 ? ? ?
p?.01 .01 ? ? ?
Design 2X4 2X3 2X2 3X3
11
6
13.2
20
4
7.4
6
9
36.0
48
FMAX(6,11) 13.2
k number of variances
2 3 4 5 6
7 8 9 10 4 23.2 37
49 59 69 79 89 97
106 5 14.9 22 28 33 38
42 46 50 54 6 11.1 15.5
19.1 22 25 27 30 32
34 7 8.89 12.1 14.5 16.5 18.4 20
22 23 24 8 7.50 9.9
11.7 13.2 14.5 15.8 16.9 17.9 18.9 9
6.54 8.5 9.9 11.1 12.1 13.1
13.9 14.7 15.3 10 5.85 7.4 8.6
9.6 10.4 11.1 11.8 12.4 12.9 12
4.91 6.1 6.9 7.6 8.2 8.7
9.1 9.5 9.9 15 4.07 4.9 5.5
6.0 6.4 6.7 7.1 7.3
7.5 20 3.32 3.8 4.3 4.6 4.9
5.1 5.3 5.5 5.6 30 2.63
3.0 3.3 3.4 3.6 3.7 3.8
3.9 4.0 60 1.96 2.2 2.3 2.4
2.4 2.5 2.5 2.6 2.6
dfFMAX
p?.01
49
Number of Means 8 ? ? ?
MSG max 18.2 26.3 34.2 18.0
MSG min 1.1 2.0 4.6 0.5
FMAX 16.5 13.2 7.4 36.0
Subjects in larger NG 10 12 21 7
dfFMAX 9 11 20 6
p?.01 .01 .01 ? ?
Design 2X4 2X3 2X2 3X3
6
4
9
50
FMAX(4,20) 7.4
k number of variances
2 3 4 5 6
7 8 9 10 4 23.2 37
49 59 69 79 89 97
106 5 14.9 22 28 33 38
42 46 50 54 6 11.1 15.5
19.1 22 25 27 30 32
34 7 8.89 12.1 14.5 16.5 18.4 20
22 23 24 8 7.50 9.9
11.7 13.2 14.5 15.8 16.9 17.9 18.9 9
6.54 8.5 9.9 11.1 12.1 13.1
13.9 14.7 15.3 10 5.85 7.4 8.6
9.6 10.4 11.1 11.8 12.4 12.9 12
4.91 6.1 6.9 7.6 8.2 8.7
9.1 9.5 9.9 15 4.07 4.9 5.5
6.0 6.4 6.7 7.1 7.3
7.5 20 3.32 3.8 4.3 4.6 4.9
5.1 5.3 5.5 5.6 30 2.63
3.0 3.3 3.4 3.6 3.7 3.8
3.9 4.0 60 1.96 2.2 2.3 2.4
2.4 2.5 2.5 2.6 2.6
dfFMAX
p?.01
51
Number of Means 8 ? ? ?
MSG max 18.2 26.3 34.2 18.0
MSG min 1.1 2.0 4.6 0.5
FMAX 16.5 13.2 7.4 36.0
Subjects in larger NG 10 12 21 7
dfFMAX 9 11 20 6
p?.01 .01 .01 .01 ?
Design 2X4 2X3 2X2 3X3
6
4
9
52
FMAX(9,6) 36.0
K number of variances
2 3 4 5 6
7 8 9 10 4 23.2 37
49 59 69 79 89 97
106 5 14.9 22 28 33 38
42 46 50 54 6 11.1 15.5
19.1 22 25 27 30 32
34 7 8.89 12.1 14.5 16.5 18.4 20
22 23 24 8 7.50 9.9
11.7 13.2 14.5 15.8 16.9 17.9 18.9 9
6.54 8.5 9.9 11.1 12.1 13.1
13.9 14.7 15.3 10 5.85 7.4 8.6
9.6 10.4 11.1 11.8 12.4 12.9 12
4.91 6.1 6.9 7.6 8.2 8.7
9.1 9.5 9.9 15 4.07 4.9 5.5
6.0 6.4 6.7 7.1 7.3
7.5 20 3.32 3.8 4.3 4.6 4.9
5.1 5.3 5.5 5.6 30 2.63
3.0 3.3 3.4 3.6 3.7 3.8
3.9 4.0 60 1.96 2.2 2.3 2.4
2.4 2.5 2.5 2.6 2.6
dfFMAX
p?.01
53
Answers to examples
Number of Means 8 ? ? ?
MSG max 18.2 26.3 34.2 18.0
MSG min 1.1 2.0 4.6 0.5
FMAX 16.5 13.2 7.4 36.0
Subjects in larger NG 10 12 21 7
dfFMAX 9 11 20 6
p?.01 .01 .01 .01 .01
Design 2X4 2X3 2X2 3X3
6
4
9
You cannot use the F test for any of
these experiments!
54
Homogeneity of Variance Conclusions
If FMAX is significant, then the assumption of
homogeneity of variance has been violated. If
the assumption of homogeneity of variance is
violated, then we cannot estimate sigma2 and
therefore can not compute the t or F test or the
Pearsons correlation coefficient (r).
55
Interval Scales
56
Assumption

• Our last assumption that we must meet to use
  parametric statistics is that the measures in our
  experiment use an interval scale.
• An interval scale is a set of numbers whose
  differences are equal at all points along the
  scale.

57
Examples of Interval Scales

• Integers - 1,2,3,4,
• Real numbers - 1.0, 1.1, 1.2, 1.3,
• Time - 1 minute, 2 minutes, 3 minutes,
• Distance - 1 foot, 2 feet, 3 feet, 4 feet,

58
Examples of Non-Interval Scales

• Ordinal - ranks, such as first, second, third
  high medium low etc.
• The difference in time between first and second
  can be very different from the time between
  second and third.
• The median is the best measure of central
  tendency for ordinal data.

59
Examples of Non-Interval Scales

• Nominal - categories, such as, male, female
  pass, fail.
• There is not even an order for nominal data.
• Categories should be mutually exclusive and
  exhaustive.
• The best measure of central tendency is the mode.

60
Comparing Scales

• Interval scales have more information than
  ordinal scales, which in turn have more
  information than nominal scales.

• The more information that is available, the more
  sensitive that a given statistical test can be.

61
Example - test grades
Interval Scale SCORES 98 84 77 76 75 62 61 60
Ordinal Scale RANKS 1 2 3 4 5 6 7 8
Nominal Scale Pass/Fail PPPPPFFF
62
Book Example - test grades
Interval Scale SCORES 98 84 77 76 75 62 61 60
Ordinal Scale RANKS 1 2 3 4 5 6 7 8
Nominal Scale Pass/Fail PPPPPFFF
Ordinal scales show the relative order of
individual measures. However, there is no
information about how far apart individuals are.
63
Book Example - test grades
Interval Scale SCORES 98 84 77 76 75 62 61 60
Ordinal Scale RANKS 1 2 3 4 5 6 7 8
Nominal Scale Pass/Fail PPPPPFFF
Categories are mutually exclusive you either
pass or fail. Categories are exhaustive you
can only pass or fail.
64
Interval Scale Conclusion

• Parametric tests can only be performed on
  interval data.
• Non-parametric tests must be used on ordinal and
  nominal data.
• Researchers prefer parametric tests because more
  information is available, which makes it easier
  to find
• Significant differences between experimental
  group means or
• Significant correlations between two variables.
• If any assumptions are violated, it is common
  practice to convert from the interval scale to
  another scale. Then you can use the weaker,
  non-parametric statistics.
• There are non-parametric statistics that
  correspond to all of the parametric statistics
  that we have studied.

65
Summary - Assumptions
To use parametric statistics, it must be true
that

• Subjects are randomly selected from the
  population.
• Experimental error is randomly distributed across
  samples in the design.
• The distribution of sample means fit a normal
  curve.
• There is homogeneity of variance demonstrated by
  using FMAX.
• The measures we take are on an interval scale.