3. parametric assumptions

Steve Saffhill

Research Methods in Sport & ExerciseBasic Statistics

Refresher

• Data handling assessment – VLE

• Basic Stats – 2 types:• Descriptives (figures & Tables) • Inferentials (accept/reject hypotheses)

• We test hypotheses with statsi.e., HO: There will be no significant correlation between attendance % and exam %

• We set an alpha (0.05) to compare to the p (probability) value SPSS gives us to make inferences about what we have found (95% Confidence)

Parametric Assumptions for Deciding on Inferential Statistics

• The aim of research is to make factual descriptive statements about a group of people. • E.g., the ingestion of creatine monohydrate over a 6 week period

before competition will enhance power output by 10%.

• However, it would be nonsense to measure every single person who might be related to this statement.

• We therefore look at a selected sample of participants and make an

educated guess about the whole of the population.

• But first you need to test if the selected sample is representative of that population

• Karl Gauss found that if you take a survey of most groups of people they will behave in certain set ways.

• It is irrelevant how you measure them, most people are average.

• There are usually some extremes at either end, but most people behave in an average, predictable way.

• This is what he called normal distribution.

• If any group of students take an exam or write a piece of coursework, most of them will score around the average (40% - 60%).

• There are always a few who score highly...

• But, to compensate, there are always a few who do badly...

Normal Distribution Curve

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Bell-shaped curve showing marks along x-axis, number of students on y-axis.

• Karl Gauss showed how most groups of people will behave in this predictable way.

• It is therefore not necessary to measure a whole population to make a true statement about it. A sample will be sufficient.

• That is.....Providing certain criteria are met, the results can then be extrapolated to the population.

• These criteria are what we call parametric properties….

= most common inferential tests for you!

What are these criteria?

• These criteria must be explored before running ANY inferential statistics!

• The descriptive statistics (last week) allow us to determine if the criteria have been met!

• If we run the wrong inferential test there is a risk of errors (type I or type II error)!

Errors in statistics• Type I = (false-positive result) occurs if the Null

hypothesis is rejected when it is actually true (e.g., the effects of the training are interpreted as being significantly different when they are not).

• Type II = (false-negative result) occurs if the Null hypothesis is accepted when it is actually false(e.g., the effects of the training are interpreted as being equal when they are actually significantly different).

So What Exactly Are these Criteria that help Us Choose the Correct Inferential Test?

• Called “the parametric assumptions”

1. Random sampling – must be randomly sampled2. Level of data being used – must be interval or ratio (high

level data).3. Normal Distribution – Must be normally distributed (2

checks: 3a and 3b)4. Equal variance in scores – The variance scores of one

variable should be twice as big as the variance of the other variable.

1. Random Sampling• Suppose I wish to make a statement about plyometric training enhancing sprint

times for all athletes.

• If I just measured a top athletics team, their sprint times might very well improve due to the best nutrition, coaching and training facilities. Plyometric training might have nothing to do with their success.

• I need to look at an unbiased sample, which I have selected by chance (i.e., randomly).

• If we have used a random sample we have accepted this 1st criteria to use parametric inferential statistics and then check the other 3 criteria

• If not: we can’t run a parametric test so run a non-parametric test (we still check the other 3 criteria for our report)

2. High Level Data• Low level data is called this because it is quite crude and

therefore hard to make good judgements about a population from a sample.

• Putting athletes in a specific order is not constructive either in terms of learning information about them.

• Supposing I told you I had been in a friendly tennis tournament and that I had come third.

• You might not be impressed……..

Until I told you that Roger Federer had come first and Kim Clijsters second

• The person who came fourth was actually the cleaner who I had persuaded to join us to make up four so the tournament could take place!!!

• You might again revise your opinion then!!!

• This is why ordinal or ranking data is low level.

Nominal/categorical - categories • E.g., male/female; rugby players/football/cricket

players; netball players/hockey players/tennis/badminton players.

Ordinal - rank order • E.g., 1st, 2nd, 3rd,

• High level data is much better to use. If I gave you a set of high level data, you could tell me a great deal about the group.

• Interval - rating scale, Fahrenheit - zero and minus are meaningful

• E.g., 28ºF, 0ºF.

• Many psychological questionnaires are interval data (eg IQ, GEQ, CSAI-2).

• Ratio - equal distances between numbers (zero means nothing, minus has no meaning) 0kg = no kilograms, -0 kg is meaningless.

• A good deal of physiological data is ratio (eg height, weight, time in minutes and seconds).

So...(after checking random sampling)

• If you have interval/ratio data we can accept the 2nd rule/criteria of parametric assumptions and we then check the other assumptions (no. 3 & 4).

• If not (i.e., we have ordinal/nominal data) we cannot run a parametric test and we must run a non-parametric test (but we still check the other 2 assumptions for our report)

3. Normal distribution

• Think back to the beginning of the lecture to Karl Gauss and how he found that if you take a survey of most groups of people, they will behave in certain set ways.

• It is irrelevant how you measure them, most people are average…some extremes but mainly average

• This is what he called normal distribution.

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Normal Distribution Curve

Bell-shaped curve showing marks along x-axis, number of students on y-axis.

• We must check whether the data does actually fit the pattern of normal distribution as the 3rd parametric rule/criteria

• It may be that if the data were plotted on a graph, it would not fit the normal pattern.

• It would then be dangerous to assume that the population behaves in the same way and run a particular test that assumes they are and lead to a type I or II error!

Normal distribution

• As the graphs we are working with are 2-D, the two ways in which the data can shift away from normality is on the horizontal axis and the vertical axis.

• It may either shift to the left extreme or right extreme. This is known as positive or negative skewness.

Resting Heart Rates of Athletes 10 Minutes After Exercise

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Resting Heart Rates of Sedentary Non-Athletes 10 Minutes After Exercise

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Or, the shift may be a very peaked or very flat curve

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Peaked = Leptokurtic

Flat = Platykurtic

Mesokurtic = normal

Normal distribution Curve• Can be visualised in SPSS when you run the descriptives:

AGE

29.028.027.026.025.024.023.022.021.0

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Std. Dev = 2.46

Mean = 25.4

N = 22.00

• While a graph is very helpful in deciding if a sample is normally distributed, it does not actually tell the researcher how skewed or kurtotic the data is.

• A number therefore needs to be found to determine accurately skewness and kurtosis values.

• These TWO numbers are found in your descriptive statistics and by dividing the skewness figure by the skewness std error.

• The answer is the skewness statistic (Z skew).

• And then repeat dividing the kurtosis figure by the kurtosis error (Z kurt).

• If these figures are between -1.96 and 1.96 it is said not to be skewed or kurtotic = Normally Distributed!!.

• These numbers are called Z scores.

Z Scores

• The information you need to work out these Z scores are provided by your descriptive statistics.

Descriptives

1.50 .0731.35

1.65

1.501.50.255.505

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.000 .343-2.089 .674

MeanLow er BoundUpper Bound

95% Conf idenceInterval for Mean

5% Trimmed MeanMedianVarianceStd. DeviationMinimumMaximumRangeInterquartile RangeSkew nessKurtosis

AnxietyStatistic Std. Error

Divide .000 by .343 to give you your Z skewness score = 0

Divide -2.089 by .674 to give you your Z kurtosis score = -3.09

Both the Z skewness and Z kurtosis scores need to fall between -1.96 and 1.96 for that variable to be deemed normally distributed

You would need to test this for ALL your variables!!!

So.....

• If all of our Z scores (skewness and kurtosis) are between -1.96 and 1.96 we meet the 3rd assumption and run a parametric test (we still check the 4th assumption but it is least important)

• If not then we must run a non-parametric test (we still check the final 4th assumption).

4. Equal (Homogeneity) Variance

• This criteria looks at two characteristics of a sample to see if their variances are reasonably similar or very different.

• You take the smallest variance, double it and see whether it is smaller or larger than the larger variance.

• If the smallest variance doubled is now larger than the biggest variance, the two data sets are known as homogenous and this criteria is accepted.

• If the smallest variance doubled is still smaller than the larger variance, the two data sets are known as heterogenous and this criteria is not accepted.

• For example, supposing our data sets have the variances 15 and 28.

• You take the smaller variance and double it. 15 x 2 = 30.

• Is 30 bigger or smaller than the biggest variance?

• Yes, 30 is bigger than 28 so the two variables are showing homogeneity of variance – so we say yes we’ve accepted this criteria and run a parametric test (pending rules 1-3)!

• For the same reason, 15 and 32 are heterogenous – therefore we wouldn’t be able to accept this criteria and run (what we do depends on rules 1-3).

The final parametric assumption….

Descri pt i ves

11. 8010 . 3001011. 1221

12. 4799

11. 735011. 7600

. 901. 9490010. 7614. 033. 27

1. 22001. 448 . 6872. 890 1. 334

15. 5680 1. 3884112. 4272

18. 7088

15. 095613. 905019. 277

4. 3905312. 5627. 0814. 52

3. 68002. 388 . 6876. 132 1. 334

MeanLower BoundUpper Bound

95% Conf idenceI nt erval f or Mean

5% Tr immed MeanMedianVar ianceSt d. Deviat ionMinimumMaximumRangeI nt erquar t ile RangeSkewnessKur t osisMean

Lower BoundUpper Bound

95% Conf idenceI nt erval f or Mean

5% Tr immed MeanMedianVar ianceSt d. Deviat ionMinimumMaximumRangeI nt erquar t ile RangeSkewnessKur t osis

GRO UPelit e

amat eur

TI MESt at ist ic St d. Error

Some inferential tests (e.g., independent t-test) double check the variance in the actual SPSS output( e.g., Levene’s test of equality of variance) as it is vital that the two groups are of equal variance for the test to run!!!

Levene’s TestInd e p e n de nt Sa mple s Te s t

8 .3 11 .0 0 4 1 .7 0 4 1 9 3 .0 9 0 .2 5 4 2 .1 4 9 1 9 -.0 4 0 0 9 .5 4 8 4 3

1 .7 2 0 1 8 5 .9 6 1 .0 8 7 .2 5 4 2 .1 4 7 7 8 -.0 3 7 3 7 .5 4 5 7 1

Eq u a l v a ria n c e sa s s u me dEq u a l v a ria n c e sn o t a s s u me d

CS1F Sig .

L e v e n e 's T e s t fo rEq u a lity o f Va ria n c e s

t d f Sig . (2 -ta ile d )Me a n

Diffe re n c eStd . Erro r

Diffe re n c e L o we r Up p e r

9 5 % Co n fid e n c eIn te rv a l o f th eDiffe re n c e

t-te s t fo r Eq u a lity o f Me a n s

• If less than .05 then there is no equal variance!

• If more than .05 then there is equal variance!

• These 4 assumptions are of progressive importance.

• If you do not meet #1 then use Non-parametric inferential tests

• Some can be violated but you must justify doing so with supporting evidence!

Some Assumptions Can be Re-Run after Re-Checking the Data

• For example if your data is not normally distributed & you have lots of cases then you could check & remove the outliers/extremes!

• Remember to justify it!! (Small v Large sample)

• You must then re-check all assumptions again

1010N =

GROUP

amateurelite

TIM

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Extreme Score

Outlier Score

You MUST go through this process every time you analyse

data!

Or…risk running the wrong tests & getting a type I or II error!