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Testing ANOVA assumptions Flashcards

1
Q

A Type I error has been defined as the

A

probability of rejecting the null hypothesis when in fact the null hypothesis is true
•This applies to every statistical test that we perform on a set of data.

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2
Q

If we perform several statistical tests on a set of data we can effectively

A

increase the chance of making a Type I error.

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3
Q

If we perform two statistical tests on the same set of data then we have a…

A

range of opportunities of making a Type I error
•Type I error on the first test only
•Type I error on the second test only
•Type I error on both the first and the second test

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4
Q

What are per comparison errors?

A

Type I errors involving single tests

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5
Q

What are familywise errors?

A

A whole set of type 1 errors

E.g. •Type I error on the first test only
•Type I error on the second test only
•Type I error on both the first and the second test

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6
Q

The relationship between pre comparison and familywise error rates is very simple:

A

Afw = c ( Apc)

Where c is the number of comparisons

  • So if we have made three comparisons, we can expect 3(0.05) = 0.15 errors. If we make twenty comparisons, we will on average make one error [200.05=1.0].
  • Of course, if we make twenty comparisons, it is possible that we may be making 0, 1, 2 or in rare cases even more errors.
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7
Q

With planned comparisons :

A

Ignore the theoretical increase in familywise type I error rates and reject the null hypothesis at the usual per comparison level.

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8
Q

With post hoc or unplanned comparisons between the means:

A

we cannot afford to ignore the increase in familywise error rate

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9
Q

A variety of different post hoc tests are commonly used - for example:

A
  • Scheffé
  • Tukey HSD
  • t-tests
  • These tests vary in their ability to protect against Type I errors.
  • Increasing Type I protection reduces Type II protection.
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10
Q

The Scheffe Test

A
  • The Scheffé is calculated in exactly the same way as a planned comparison
  • Scheffé differs in terms of the FCritical that is adopted.
  • For the one-way between groups analysis of variance the critical F associated with an FScheffé is given by:
  • where a is the number of treatment levels and F(dfA, dfS/A) is the critical value of F for the overall, omnibus analysis of variance.
  • For our example
  • Omnibus ANOVA critical value F(2,12)= 3.885. There were three treatment levels so (3-1)*3.885= 7.77.
  • Fobserved = 14.29
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11
Q

Tukey HSD

A
  • The Tukey (Honestly Significant Difference) test establishes a value for the smallest possible significant difference between two means.
  • Any mean difference greater than the critical difference is significant
  • The critical difference is given by:
  • where q(a,df,a) is found in tables of the studentized range.
  • This particular formula only works for between groups analysis of variance with equal cell sizes
  • A variety of different formulae are used for different designs
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12
Q

What is a Bonferroni correction?

A

When comparing two means, a modified form of the t-test is available.

  • For multiple comparisons the critical value of t is found using
  • p=0.05/c
  • where c is the number of comparisons.
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13
Q

When comparing two means, a:

A

modified form of the t-test is available

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14
Q

For multiple comparisons the critical value of t is found using:

A
  • p=0.05/c

* where c is the number of comparisons

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15
Q

Post Hoc Tests

A
  • Post-hoc tests are conservative – they reduce the chance of type I errors by greatly increasing type II errors.
  • Only very robust effects will be significant.
  • Null results using these tests are not easy to interpret.
  • Many different post hoc tests exists and have different merits and problems
  • Many post hoc tests are available on computer based statistical packages (e.g. SPSS or Experstat)
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16
Q

Post Hoc tests are conservative, this means…

A
  • they reduce the chance of type I errors by greatly increasing type II errors.
  • Only very robust effects will be significant.
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17
Q

Null results using post Hoc tests are not…

A

Easy to interpret
•Many different post hoc tests exists and have different merits and problems
•Many post hoc tests are available on computer based statistical packages (e.g. SPSS or Experstat)

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18
Q

Many post hoc tests are available on

A

computer based statistical packages (e.g. SPSS or Experstat)

19
Q

The assumptions of the F ratio

A
  • Independence
  • The numerator and denominator of the F-ratio are independent
  • Random Sampling
  • Observations are random samples from the populations
  • Homogeneity of Variance
  • The different treatment populations have the same variance.
  • Normality
  • Observations are drawn from normally distributed populations
20
Q

The assumptions of the F ratio:

A

Independence
Random sampling
Homogeneity of variance
Normality

21
Q

The assumptions of the F ratio

Independence

A

The numerator and denominator of the F-ratio are independent

22
Q

The assumptions of the F ratio

Random Sampling

A

Observations are random samples from the populations

23
Q

The assumptions of the F ratio

Homogeneity of Variance

A

The different treatment populations have the same variance

24
Q

The assumptions of the F ratio

Normality

A

•Observations are drawn from normally distributed populations

25
Q

Testing assumptions of ANOVA

Each of these assumption should be met before…

A

progressing onto the analysis.

26
Q

There are two assumptions of ANOVA that we have to assume have been met by the experimenter…

A
  • Independence and Random Sampling

* If an experiment has been designed appropriately both of these assumptions will be true.

27
Q

Both the homogeneity of variance and the normality assumptions of ANOVA…

A

need not necessarily be true

28
Q

Testing Homogeneity of variance

When looking a between groups designs use

A
  • Hartley’s F-max
  • Bartlett
  • Cochran’s C
  • All these tests are sensitive to departures from normality
  • All of these tests are available in SPSS (as are a number of other tests)
29
Q

Testing homogeneity of variance

When looking at within or mixed designs use

A

Box’s M

30
Q

For hand calculations, there is a quick and dirty measure of homogeneity of variance:

A

Largest variance / smallest variance

Smaller than 4?
this is a heuristic. When you have the option, use one of the specific tests (e.g. Bartlett).

31
Q

•The three most commonly used tests for normality are:

A
  • Skew
  • Lilliefors
  • Shapiro-Wilks
  • These tests compare the distribution of the data to a theoretically derived normal distribution.
  • All these tests are very sensitive to departures from normality when there are large samples.
32
Q

The Lilliefors and Shapiro-Wilks are difficult to

A

calculate by hand, but both are available on SPSS.

33
Q

Testing normality by examining skew

A
  • Since we assume
  • that the distributions of the population from which the samples are taken are normal
  • and the skew of a normal distribution is equal to zero
  • Then
  • One test of normality is to see if the skew is significantly different to zero
  • In other words, test the value of skew to see if it deviates significantly from a normal distribution.
34
Q

The simplest test we can use is ____ to test for skew

A

Z-score

35
Q

Transforming data reduces the probability of making a

A

type II error

36
Q

type II error occurs when we fail to

A

reject the null hypothesis when it is false

37
Q

If an assumption is broken, ANOVA fails gracefully:

A

we will miss real effects (type II) but we will not increase our rate of making claiming effects that do not exist (type I)

38
Q

Data should be transformed when either:

A

the data is not homogenous or not normal

39
Q

Solving the homogeneity problem often solves the:

A

normality problem and vice versa

40
Q

What happens when transforming the data is impossible?

A

•In general we proceed with the analysis but advise caution to the reader when reporting the results
•This is particularly important if the observed F value has an associated probability, p, such that
0.1>p>0.01
•In these circumstances it is difficult to know whether a type I error or a type II error is being made or if no error is being made at all.

41
Q

What can we do in order to meet the assumption of the analysis of variance?

A
  • In order to return our data to normality and establish homogeneity of variance we can use transformations.
  • These are simply mathematical operations that are applied to the data before we conduct an analysis of variance.
  • However, there are three circumstances where no transformation to the data will work:
  • Variances are heterogenous
  • Distributions are heterogenous
  • Variances are heterogeneous and distributions are heterogeneous
42
Q

In the case of skew the z-score is given by:

A

Z = skew-0 / SE skew

43
Q

The standard error of skew is given by:

A

SEskew = square root of 6/N

  • where N is the number of cases in the sample.
  • If a z score associated with the skew is greater than |±1.96| then the sample is significantly different from normal.
  • In other words, a value of skew which is significantly different from zero, would mean that we do not have normally distributed data

Stats (19 decks)

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