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Statistics using Parametric and non-parametric tests.www.busmgt.ulst.ac.uk/modules/sls701j1/stats/numtests.

Parametric and Non-Parametric tests of numerical variables

When selecting a statistical test we must ask three questions.

Are we using independent orrelated samples?

Are we comparing two samples or more than two samples?

Do the variables satisfy the assumptions of parametric statistics?

Independent or Related Samples

Where we are comparing different groups of people, we are comparing independent samples.For example males and females ordefenders, midfielders and forwards.

Where we are comparing the same group of people but under different circumstances, we

have related (or matched) samples. For example before and after or when taking glucose,when taking a placebo and when taking nothing.

Two or more than two samples

We may wish to compare two independent samples; eg males and females.

We may wish to compare two related samples; eg performance before and performance after.

We may wish to compare more than two independent samples; eg defenders, midfielders andforwards.

We may wish to compare more than two related samples; eg performance when takingglucose, performance when taking a placebo and performance when taking nothing.

The Assumptions of Parametric Statistics There are three assumptions that must besatisfied in order to validly apply parametric statistical tests.

The dependent variable must be measured on an interval or ratio scale.

The data must be normally distributed.

The variances of the different samples must be homogeneous.

If any of these assumptions are violated, non-parametric procedures should be used. Thefollowing table shows the different tests available and an indication of which should be

selected.

Parametric Non-parametric

Two independent samples Independent t-test Mann Whitney U test

Two related samples Paired t-test Wilcoxon Signed Ranks test

3+ independent samples One-way ANOVA Kruskall Wallis H test

3+ related samples Repeated measures ANOVA Friedman two-way ANOVA

The Assumptions of Parametric Satistics

In order to apply parametric statistical procedures, the following three assumptions must hold.
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The dependent variables must be measured on an interval or ratio scale.

The variables must be normally distributed.

The variances of the variables must be homogeneous.

If any of these assumptions are not valid then non-parametric procedures should be applied.

Testing the Normality of Variables

In order to apply parametric statistical procedures, variables must be normally distributed withsimilar numbers of cases on either side of the mean and approximately 68% of the caseswithin one standard deviation of the mean. A varaible will not be normally distributed if it isany of the following

Positively skewed - there is a wider variation above the modal range of values thanbelow it.

Negatively skewed - there is a wider variation below the modal range of values thanabove it.

Lepokurtic - too narrowly dispersed about the mean. Platykurtic - too widely dispersed about the mean.

Is SPSS, we can investigate skewness and kurtosis either through the descriptives facilityor though the explore facility of the descriptive statistics submenu of the analyse menu.Where skewness and kurtosis are to be included in the output, each is represented by aStatistic and a SD.

If we divide the Skewness statistic by its SD we get a value for zSkew which must be in therange -2 to +2 or else the variable is skewed (less than -2 meaning negatively skewed, morethan +2 meaning positively skewed).

Similarly, if we divide the Kurtosis statistic by its SD we get a value for z Kurt which must be inthe range -2 to +2 or else the variable is not mesokurtic (less than -2 meaning platykurtic,more than +2 meaning lepokurtic).

Testing the Homogeneity of Variables

Levene's test of Equality of Variances is used with the Independent t-Testand some of theANOVA tests. It is possible to proceed with the independent t-Test even if the variances arenot equal by reducing the number of degrees of freedom. The output of an independent t-testin SPSS contain two rows of alternative results; the first can be used if equal variances canbe assumed and the second can be used if equal variances cannot be assumed.

Non-Parametric tests of numerical variables

Non-parametric tests are used when the assumptions necessary for parametric proceduresdo not apply. The non-parametric procedures are "distribution free" tests that do not usesample parameters (mean and standard deviation). Instead, the tests transform all of thevalues into ranks. Therefore, non-parametric tests are not considered to be as case-sensitiveor powerful as theirparametric equivalents.

The Independent t-Test

The purpose of the independent t-test is to compare two independnet groups of subjects interms of some dependent variable. In order to apply the indepenendent t-test, the

assumptions of parametric statisticsshould hold. If any of these assumptions are violated,then the Mann-Whitney U testshould be used.
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The significance of the difference between the two groups depends on three things

The difference between the two group means.

The variation about the mean of each group.

The number of subjects in each group.

An Example

Consider a two groups of 24 soccer players distinguished by their level of competition; anelite group and a semi-professionalgroup. The independent t-test can be used to determineif the distance covered during a soccer match is significantly different between these twogroups. The following section uses this example and an SPSS datasheet containing thedetails of the 48 soccer players is provided.

Performing an Independent t-Test in SPSS

From the Analyse menu select the Compare Means sub-menu and from that select the

Independent Samples t-Test option. The Independent Samples t-Test popup window willappear. Transfer the independent variable to "Grouping Variable" and the dependent variableto "Test variable(s)". Note that more than one test variable means that a series ofindependent t-tests can be performed at the same time. Click on the Define Groups button;this will allow you to identify the values of the grouping variable that identify the twoindependent groups being compared. Click on Continue to remove the define groups popupwindow and then click on the OK button; the results will appear shortly.

The first table presents descriptive statistics for the two groups in terms of the test variable.The second table presents two rows of alternative results for the independent t-test. The firstrow should be used if the variances of the two groups are homogeneous, otherwise thesecond row of results should be used. The homogeneity of the variances of the two groups istested by Levene's test for Equality of Variances. If the significance of this test is less than

0.05 then we must assume unequal variances and use the second row of results.

Whichever row of results are used, the most important columns are

t - the test score.

df - the number of degrees of freedom.

Sig (2 tailed) - the P value.

Doing the example in SPSS

The variable "level" has two values; 1 for elite players and 2 for semi-professional players.

The variable "distance" is the test variable.

From the Analyse menu select the Compare Means sub-menu and from that select theIndependent Samples t-Test option. The Independent Samples t-Test popup window willappear. Transfer level to "Grouping Variable" and distance to "Test variable(s)". Click on theDefine Groups button; this will allow you to identify the values of the grouping variable thatidentify the two independent groups being compared; in this case 1 for elite players and 2 forsemi-professionals. Click on Continue to remove the define groups popup window and thenclick on the OK button; the results will appear shortly.

The first table presents descriptive statistics for the two groups in terms of distance covered.We can see that the mean distance covered for the elite players (9262.46 m) is greater thanthe mean distance covered for the semi-professionals (8795.58 m). As Levene's test forEquality of Variances has not shown a significant difference between the two variances (P >0.780), we assume equal variances and use the top row of independent t-test results.
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Examining the t-test results, we can see that t with 46 degrees of freedom = 1.5 and that P isgreater than 0.05. The results could be reported as follows

The distance covered by elite players was not significantly greater than that covered by semi-professional players (9262.5+1101.2 m v 8795.6+1021.9 m, t46 = 1.5, P > 0.05).

Do the test using the SPSS datasheet provided and examine the output produced to seewhere the details in the above results are extracted from.

SPSS datasheet for this example

Exercise

Consider two groups of 14 netball players distinguished by level; international and clubplayers. The distance covered by these players is measured during a netball match. Use anindependent t-test to determine if level of player (level; 1 = international, 2 = club) significantlyinfluences the distance covered during competition (distance).

Click here to access datasheet for this exercise

Click here to view the solution

The Mann-Whitney U Test

The Mann-Whitney U test is the non-parametric alternative to the Independent t-Test. TheMann-Whitney U test should be used to compare two independent groups of subjects in termsof some dependent variable if theassumptions of parametric statistics do not hold.

There are some assumptiions that must hold when applying the Mann-Whitney U test, namely

Independent random samples must have been used.

The scale of measurement is at least ordinal.

The Mann-Whitney U test works by transforming all of the values into ranks and thensumming the ranks of each of the two independent group of values. A z-score is determined,to express the difference between the sum of ranks and the expected sum of ranks if the twogroups were evenly interleaved in a normalised form.

An Example

Consider a two groups of 24 soccer players distinguished by their level of competition; anelite group and a semi-professional group. The Mann-Whitney U test can be used todetermine if the distance covered during a soccer match is significantly different betweenthese two groups. The following section uses this example and an SPSS datasheetcontaining the details of the 48 soccer players is provided.

Performing an Mann-Whitney U Test in SPSS

From the Analyse menu select the Non-Parametric tests sub-menu and from that select the2 Independent Samples option. The Two Independent Samples Test popup window willappear. Transfer the independent variable to "Grouping Variable" and the dependent variableto "Test variable(s)". Note that more than one test variable means that a series of Mann-Whitney U tests can be performed at the same time. Click on the Define Groups button; this
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will allow you to identify the values of the grouping variable that identify the two independentgroups being compared. Click on Continue to remove the define groups popup window andthen click on the OK button; the results will appear shortly.

The most important columns are

z - the test score.



The variable "level" has two values; 1 for elite players and 2 for semi-professional players.The variable "distance" is the test variable.

From the Analyse menu select the Non-Parametric tests sub-menu and from that select the2 Independent Samples option. The Two Independent Samples Test popup window willappear. Transfer level to "Grouping Variable" and distance to "Test variable(s)". Click on the

Define Groups button; this will allow you to identify the values of the grouping variable thatidentify the two independent groups being compared; in this case 1 for elite players and 2 forsemi-professional players. Click on Continue to remove the define groups popup windowand then click on the OK button; the results will appear shortly.

The first table presents mean rank and sum of ranks for the distance covered by members ofthe two groups. Examining the test results, we can see that the z-score of -1.516 is notsignificant as the P value of 0.130 is greater than 0.05. The results could be reported asfollows

A Mann-Whitney U test revealed that the distance covered by elite players was notsignificantly greater than that covered by semi-professional players (z = -1.5, P > 0.05).



Exercise

Consider two groups of 14 netball players distinguished by level; international and clubplayers. The distance covered by these players is measured during a netball match. Use aMann-Whitney U test to determine if level of player (level; 1 = international, 2 = club)significantly influences the distance covered during competition (distance).



The Paired t-Test
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The purpose of the paired t-test is to compare two related samples. When using relatedsamples, there is a single group of subjects with a concept of interest being measured onmore than one occasion or under more than one circumstance. For example, we may wish tocompare the distance covered by a group of footballers in the first half with the distancecovered in the second half. In order to apply the paired t-test, the assumptions of parametric

statistics should hold. If any of these assumptions are violated, then the Wilcoxon SignedRanks test should be used.

Unlike the independent t-test, the large variation in the related samples will not necessarilyprevent a difference in the means from being significant. Consider a sample of 100 people ofa wide range of weights who are placed on a six week exercise programme. We wish tocompare their weight before and after the exercise programme. If the exercise trainingprogramme produces a consistent reduction in weight over the six week period, the paired t-test should conclude that there is a significant difference, even if weight of the group of 100people had a large variation about the mean before and after.

An Example

Consider a group of 48 soccer players whose distance covered during a soccer match ismeasured during the first half and the second half. The paired t-test can be used to determineif the distance covered is significantly different between the first and second halves. Thefollowing section uses this example and an SPSS datasheet containing the details of the 48soccer players is provided.

Performing an Paired t-Test in SPSS

From the Analyse menu select the Compare Means sub-menu and from that select thePaired Samples t-Test option. The Paired Samples t-Test popup window will appear. Selectthe two variables representing the quantity of interest under the two different circumstances

(eg before and after). Note that the arrow to transfer these does not become available untiltwo variables are highlighted. Transfer these to the variables to be analysed. Click on the OKbutton; the results will appear shortly.

The first table presents descriptive statistics for the two related samples. The second tableshows the correlation between the two variables. The third table provides the results of thepaired t-test. The most important columns are

t - the test score.

df - the number of degrees of freedom.



The variables "first" and "second" represent the distance covered in the first and secondhalves respectively.

From the Analyse menu select the Compare Means sub-menu and from that select thePaired Samples t-Test option. The Paired Samples t-Test popup window will appear. Selectfirst and second and transfer these to the variables to be analysed. Click on the OK button;the results will appear shortly.

The first table presents descriptive statistics. We can see that the mean distance covered is4675.10 m in the first half and 4353.92 m in the second half. Examining the paired t-test

results in the third table, we can see that t with 47 degrees of freedom = 4.8 and that the Pvalue of 0.000 is less than 0.05. Note that a P value of 0.000 is only to 3 decimal places so
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we should report P < 0.001. The results could be reported as follows

The soccer players covered a significantly greater distance in the first half than the second half(4675.1+596.1 m v 4353.92+575.0 m, t47 = 4.8, P < 0.001).



Exercise

Consider a group of 28 netball players whose heart rate is recorded during a match as well asduring intervals between quarters. Use a paired t-test to determine if the heart rate recordedduring match time (hr_match) is significantly different to the heart rate recorded duringintervals (hr_int).



ANOVA models

This area of POD's Guide to statistics is not yet completed but when it is completed forOctober 2000, it will provide some general information on univariate ANOVA tests and coverthe statistical tests.

There will be information on

The general assumptions for ANOVA tests

Between and within subjects factors

The F-ratio

The different ANOVA models

One-way ANOVA

Repeated measures ANOVA

Factorial ANOVA with between subjects factors

Factorial ANOVA with within subjects factors (the within-within design)

Factorial ANOVA combining between and within subjects factors (the mixed ANOVA)

ANCOVA (analysis of covariance)

The One-way ANOVA

The purpose of the one way ANOVA is to compare more than two independent samples. Forexample, we may wish to compare the distance covered by four groups of footballers; centre-

backs, wing-backs, midfielders and forwards. In order to apply the one-way ANOVA, the
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The first table of results shows descriptive statistics; the midfielders cover the largestdistance (9796.88 m) followed by wing-backs (9053.63 m), forwards (8555.58 m) and centre-backs (8462.25 m).

The second table reports on Levene's test of the homogeneity of variances. The P valueof 0.308 is greater than 0.05 showing that there is no significant difference between the

variances of the four groups. If it had been less than 0.05, our use of the one-way ANOVAwould have been invalid.

The third table provides the ANOVA results. The F-ratio (6.082), its significance (0.001) andthe degrees of freedom (3 for between groups and 44 for within groups) are particularlyimportant.

The fourth table provides the results of the LSD post hoc tests. The "Sig" column of thistable shows the significance of the difference between each pair of positions for the distancecovered during competition. The results should be reported as follows.

A one-way ANOVA revealed that positional role significantly influenced the distance covered

during competition (F3,44 = 6.1, P < 0.01). Fisher LSD post hoc tests revealed that the9796.9+1094.5 m was significantly greater than the 8462.3+781.9 m covered by centre-backs(P < 0.01) and the 8555.6+767.6 m covered by forwards.



Exercise

Consider seven groups of 4 netball players distinguished by positional role. The distance

covered by these players is measured during a netball match. Use a one-way ANOVA todetermine if positional role (pos) significantly influences the distance covered duringcompetition (distance). Use LSD post hoc tests to explore differences between individualpairs of positions.



The Kruskall Wallis H Test

The Kruskall Wallis H test is the non-parametric alternative to the One-way ANOVA. TheKruskall Wallis H test should be used to compare three or more independent samples if theassumptions of parametric statistics do not hold. We might use this test to compare thedistance covered by players of four different positional groups (centre-backs, wing-backs,midfielders and forwards) during a football match.

The Kruskall Wallis H test works by transforming the values into ranks before determining thesum of the ranks for each group. Where there are five or more subjects in each group, the

test approximates to the 2 statistic. The number of degrees of freedom used with the chi-squared test is one less than the number of groups involved.

An Example
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Consider a four groups of soccer players distinguished by their positional role; centre-backs(n=12), wing-backs (n=8), midfielders (n=16) and forwards (n=12). The Kruskall Wallis H testcan be used to determine if the distance covered during a soccer match is significantlyinfluenced by positional role. The following section uses this example and an SPSSdatasheet containing the details of the 48 soccer players is provided.

Performing a Kruskall Wallis H Test in SPSS

From the Analyse menu select the Non-Parametric tests sub-menu and from that select theK Independent Samples option. The Test for Several Independent Samples popup windowwill appear. Transfer the dependent variable of interest to the area labelled "Test VariableList" and transfer the variable that distinguishes the different groups of subjects to the arealabelled "Grouping Variable". Click on the Define Range button and enter the values of thegrouping variable that identify the first and last group. Click on continue to remove thispopup and then OK to see view the results.


2 - the test score. df - the degrees of freedom.



The variable "pos" has four values; 1 for centre-backs, 2 for wing-backs, 3 for midfielders and4 for forwards.The variable "distance" is the test variable.

From the Analyse menu select the Non-Parametric tests sub-menu and from that select theK Independent Samples option. The Test for Several Independent Samples popup windowwill appear. Transfer distance to the area labelled "Test Variable List" and transfer pos to thearea labelled "Grouping Variable". Click on the Define Range button and enter the values ofthe grouping variable that identify the first and last group; in this case 1 and 4 for centr-backsand forwards respectively. Click on continue to remove this popup and then OK to see viewthe results.

The first table provides the mean rank of each position. The mean rank for centre-backs is16.58 compared with 34.19 for midfielders. This indicates that centre-backs cover the leastdistance and midfielders cover the most distance.

The second table provides the test results. The important figures are the chi-squared teststatistic of 13.506, the number of degrees of freedom which is 3 and the P value of 0.004.

The results should be reported as follows.

A Kruskall Wallis H test revealed that positional role significantly influenced the distance

covered during competition ( 23 = 13.5, P < 0.01).



Exercise

Consider seven groups of 4 netball players distinguished by positional role. The distancecovered by these players is measured during a netball match. Use a Kruskall Wallis H test to
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determine if positional role (pos) significantly influences the distance covered duringcompetition (distance).

Note - where there are five or more subjects in each group, the Kruskall Wallis H testapproximates to the chi squared distribution. Therefore, this example is not a truely valid useof the test.



The Repeated Measures ANOVA

The purpose of the Repeated Measures ANOVA is to compare more than two relatedsamples. For example, we may wish to compare the body weight of a group of subjects onseveral occasions over the course of an exercise training programme. In order to apply theRepeated Measures ANOVA, the assumptions of parametric statistics should hold. If any ofthese assumptions are violated, then the Friedman Two-Way ANOVAshould be used.

The Repeated Measures ANOVA, like all other ANOVA tests, uses the F-ratio. The F-ratiorepresents the ratio of the variation of subjects when measured on different occasions withthe variation among subjects measured on the same occasion.

The result of the Repeated Measures ANOVA indicates whether there is a difference in ameasurement between the different occasions it is measured. To explore differencesbetween individual pairs of occasions, it is necessary to employ post hoc tests.

The Assumption of Sphericity

In addition to the assumptions of parametric statistics, the Repaeated Measures ANOVA hasan additional assumption, the assumption of sphericity. Sphericity requires that repeatedmeasures have both homogeneity of variance and homogeneity of covariance. Homogeneityof covariance means that the repeated measures must have equal correlations.

The results of the Repeated Measures ANOVA should be preceeded by the results ofMauchly's test of Sphericity. A significant result of Mauchly's test means that the assumptionof Sphericity must be rejected. If this is the case, the probability of making a type I errorincreases.

However, there are methods for correcting for violoation of Sphericity by adjusting thedegrees of freedom being used. There is the Greenhouse-Geisser (GG) adjustment and the

Huynh-Feldt (HF) adjustment. Each of these adjustments involves scaling the number ofdegrees of freedom using an epsilon value.

The strategy for addressing the assumption of sphericity when reading the output provided bySPSS is summarised in the following steps

If Mauchly's test of Sphericity returns a non-significant result then use the rows ofresults entitled "Sphericity Assumed" and don't worry about the following steps.

If the F-ratio using the Greenhouse-Geisser (GG) adjustment is significant thenconclude that the within subjects effect has a significant influence on the dependentvariable. Report the GG results including epsilon.

If the F-ratio using the Greenhouse-Geisser (GG) adjustment is not significant then

use the row of results entitled "Sphericity Assumed".
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If the F-ratio using the Greenhouse-Geisser (GG) adjustment is not significant, butthe F-ratio with no adjustment is significant, use the Huynh-Feldt (HF) adjustment.Report the HF results including epsilon.

An Example

Consider a group of 28 netball players whose distance covered during competition ismeasured during each 15 minute quarter of the match. The Repeated Measures ANOVA canbe used to determine if the distance covered is significantly different between differentquarters of the match. If there is a significant between the quarters, then post hoc tests canbe employed to explore the significance of differences between individual pairs of quarters.The following section uses this example and an SPSS datasheet containing the details of the28 netball players is provided.

Performing a Repeated Measures ANOVA in SPSS

From the Analyse menu select the General Linear Model sub-menu and from that select the

GLM Repeated Measures option. The GLM Repeated Measures popup window will appear.If you have measured a concept on several occasions, you will not have a single variablerepresenting that concept; there will be several variables representing that concept. Youmust, therefore, provide a single name for the concept in the area labelled "Within SubjectFactor Name". You must then enter the number of occasions on which the concept wasmeasured. This is placed in the area labelled "Number of Levels". This within subjects factoris then included by clicking on Add. SPSS has no way of knowing which of your variablesrepresent each measurement of the concept. Therefore, you must click on Define whichpresents you with a further popup window where you identify the variable representing eachrepeated measurment of the concept.

To perform post hoc tests, do not use the Post Hoc facility provided! This only allows posthoc tests to be applied to between-subjects factors. Post hoc tests are performed through the

Options facility.

Click on the OK button to view the results. The ANOVA table provides three rows of resultsresults. The most important figures are

F - the F-ratio.

Sig - the P value.

df from the first row - the degrees of freedom for the main effect (between groups).

df from the second row - the error degrees of freedom (within groups).


The variables "dist_q1", "dist_q2", "dist_q3" and "dist_q4" represent the distance coveredduring the four quarters of a netball match.

From the Analyse menu select the General Linear Model sub-menu and from that select theGLM Repeated Measures option. The GLM Repeated Measures popup window will appear.Enter "quarter" as a name for the concept in the area labelled "Within Subject Factor Name".Enter 4 into the area labelled "Number of Levels". This within subjects factor is then includedby clicking on Add. SPSS has no way of knowing which of your variables represent eachmeasurement of the concept. Therefore, you must click on Define which presents you with afurther popup window where you identify the variable representing each repeatedmeasurment of the concept; "dist_q1", "dist_q2", "dist_q3" and "dist_q4". To perform post hoctests, use the Options facility. Within the options popup window, transfer Quarter to the area

labelled "Display means for" and click on "Compare main effect". There is a choice of threepost hoc tests; LSD, Bonferroni and Sidak. In this example use LSD. Click on the OK buttonto view the results.


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The first thing to note is that Mauchly's test reveals that sphericity cannot be assumed (P =0.004). The F-ratio of 7.733 is significant (P = 0.001) using the Greenhouse-Geisser (GG)adjustment. Therefore, we conclude that the distance covered by netball players issignificantly different between the four quarters of a match. The Repeated Measures ANOVAresults are extracted from the Greenhouse-Geisser (GG) row of the "tests of within subjectseffects" table on this occasion, reporting epsilon. The post hoc test results should follow. The

results could be reported as follows.

A repeated measures ANOVA revealed that distance covered was significantly differentbetween the four quarters of a netball match (F 3,81 = 7.3, GG epsilon = 0.707, P < 0.01).Fisher LSD post hoc tests revealed that the distance covered during the fourth quarter wassignificantly less than that covered in the first (P < 0.01), second (P < 0.01) and third quarters(P < 0.05) and that the distance covered in the third quarter was significantly less than thethat covered in the first quarter (P < 0.05).



Exercise

Consider a group of 28 netball players whose %heart rate max is recorded during the fourquarters of a netball match; these are represented by the variables hr_q1, hr_q2, hr_q3 andhr_q4. Use a repeated measures ANOVA to determine if heart rate significantly differsbetween the four quarters. Use LSD post hoc tests to explore differences between individualpairs of quarters.

Click here to access datasheet for this exercise (this is the same data file as in the workedexample - you may alreadu have it open in SPSS).


The Friedman Two-Way ANOVA Test

The Friedman two-way ANOVA is the non-parametric alternative to the Repeated MeasuresANOVA. The Friedman two-way ANOVA should be used to compare three or more relatedsamples if the assumptions of parametric statistics do not hold. We might use this test tocompare the distance covered by 28 players during four different quarters of a netball match.

The Friedman two-way ANOVA works by grouping the values into a number of orderedgroups equivalent to the number of related samples being compared; in the netball example

this is four groups of values. If there is no difference between the four quarters, then wewould expect the frequency of ordered group membership for the values of each quarter to besimilar. This is tested using a chi squared testwith a number of degrees of freedom beingone less than the number of related samples involved.

An Example

Consider a group of 28 netball players whose distance covered during competition ismeasured during each 15 minute quarter of the match. The Friedman two-way ANOVA canbe used to determine if the distance covered is significantly different between differentquarters of the match. The following section uses this example and an SPSS datasheetcontaining the details of the 28 netball players is provided.

Performing a Friedman Two-Way ANOVA Test in SPSS
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From the Analyse menu select the Non-Parametric tests sub-menu and from that select theK Related Samples option. The Test for Several Related Samples popup window willappear. The concept of interest is represented by several variables; one for each relatedsample. Each of these variables must be transferred to the area labelled "Test Variables".Click on OK to view the results.


2 - the test score.

df - the degrees of freedom.



The variables "dist_q1", "dist_q2", "dist_q3" and "dist_q4" represent the distance coveredduring the four quarters of a netball match.

From the Analyse menu select the Non-Parametric tests sub-menu and from that select theK Related Samples option. The Test for Several Related Samples popup window willappear. The variables "dist_q1", "dist_q2", "dist_q3" and "dist_q4" must be transferred to thearea labelled "Test Variables". Click on OK to view the results. The results could be reportedas follows.

A Friedman two-way ANOVA revealed that distance covered was significantly different

between the four quarters of a netball match ( 23 = 13.3, P < 0.01). Less distance wascovered in the fourth quarter than in any of the preceding quarters.



Exercise

Consider a group of 28 netball players whose %heart rate max is recorded during the fourquarters of a netball match; these are represented by the variables hr_q1, hr_q2, hr_q3 andhr_q4. Use a Friedman two-way ANOVA to determine if heart rate significantly differsbetween the four quarters.

Click here to access datasheet for this exercise (this is the same data file as in the workedexample - you may alreadu have it open in SPSS).


Factorial ANOVA with between subjects factors

The purpose of the factorial ANOVA is to determine the influence of more than one factor onsome dependent variable as well as the interaction of those factors. A between-subjectsfactor is used to distinguish independent groups of subjects. For example, we may wish todetermine the influence of level and position on the distance footballers. In order to apply afactorial ANOVA, the assumptions of parametric statisticsshould hold.

The factorial ANOVA produces an F-ratio for each factor as well as each interaction between

factors. In the case of a two-way factorial ANOVA that includes level and position as betweensubjects factors, there will be three F-ratio produced
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An F-ration for level

An F-ratio for position

An F-ratio for the interaction of level and position

The F-ratio represents the ratio of the variation of players of different groups to the variation ofplayers of the same group.

The results of the factorial ANOVA firstly indicate whether each factor has a significantinfluence on the dependent variable. It is only necessary to use post-hoc tests for thosesignificant factors that are measured at more than two levels. Therefore, post-hoc testsshould be used to compare different positions but not to compare different levels. There areonly two levels; elite and semi-professional. If the F-ratio for level is significant, the differenceis obviously between the only two groups involved.

The results secondly indicate whether there is a significant interaction between factors. Aninteraction occurs where the pattern of distances covered by players of different positionsdiffers between players of different levels. This means that distance covered is influenced bythe combination of level and position.

An Example

Consider a group of 48 soccer players that are distinguished by level and position. There aretwo levels (elite and semi-professional) and four positions (centr-back, wing-back, midfieldand forward). The factorial ANOVA can be used to determine if the distance covered during asoccer match is significantly influenced by level, position as well as the interaction betweenthese two factors. If there is a significant between positions, then post hoc tests can beemployed to explore the significance of differences between individual pairs of positions. If,however, level has a significant influence on distance covered, post-hoc tests are not requiredas this factor is only measured at two levels.

Performing a Factorial ANOVA with between subjects factors inSPSS

From the Analyse menu select the General Linear Model sub-menu and from that select theUnivariate option. The GLM - General Factorial popup window will appear. The variablesthat identify the factors of interest should be transferred to either "Fixed Factor(s)" or"Random factor(s)" and the dependent variable should be transferred to "DependentVariable". The difference between a fixed factor and a random factor is that the for randomfactors the subjects would be selected at random. The Post Hoc button allows the results ofpost hoc tests to be included in the output. Note, however, that post hoc tests for randomfactor(s) are achieved through Options. The Options button provides a popup window thatallows the user to have descriptive statistics for the different samples included in the results.

Other options allow Levene's test of homogeneity of variances to be done. There are varioustables of output provided. The table entitled "Tests of Between Subjects Effects" shows theF-ratio and significance of each factor as well as each factor interaction. The degrees offreedoem for each factor are important as are the error degrees of freedom.


The variable "level" has two values; 1 for elite and 2 for semi-professional.The variable "pos" has four values; 1 for centre-backs, 2 for wing-backs, 3 for midfielders and4 for forwards.The variable "distance" is the dependent variable.

From the Analyse menu select the General Linear Model sub-menu and from that select theGLM - General Factorial option. The GLM - General Factorial popup window will appear.
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The variables level and pos should be transferred to "Random Factor(s)" and distance shouldbe transferred to "Dependent List". The Options button provides a popup window that allowsthe user to include descriptive statistics for the different positions in the results. The Optionsbutton is also used to compare pos using Fisher's Least Significant Difference post hoc test(LSD).

Note the table of descriptive statistics. For the elite subjects the midfielders cover the largestdistance (10227.38 m) followed by wing-backs (9200.25 m), centre-backs (8746.50 m) andforwards (8533.33 m). For the semi-professional subjects the midfielders cover the largestdistance (9366.38 m) followed by wing-backs (8907.00 m), forwards (8577.33) and centre-backs (8178.00 m).

The ANOVA results reveal thatposition has a significant influence on distance covered duringcompetition (F3,3 = 10.676, P < 0.05). LSD post hoc tests reveal that midfielders coveredsignificantly more distance than centre-backs (P < 0.01) and forwards (P < 0.01). However,neither level (F1,3.458 = 3.869, P > 0.05) nore the interaction of level and position (F3,40 = 0.580,P > 0.05) had a significnat influence on distance covered.



Exercise

Try the above example again but this time include level and position as fixed rather thanrandom factors.

Click here to access datasheet for this exercise (note this datasheet may already be open asit is the same data as for the previous example).


Factorial ANOVA with between and within subjects factors - themixed ANOVA

The purpose of the factorial ANOVA is to determine the influence of more than one factor onsome dependent variable as well as the interaction of those factors. A between-subjectsfactor is used to distinguish independent groups of subjects. Positional role is a betweensubjects factor as it distinguishes independent groups of footballers. A within-subjects factorinvolves samples related to the same group of subjects. For example the halves of a footballare not participated in by independent groups of subjects, they all participate in both halves.We may wish to determine the influence of position on the distance covered by footballers. Inorder to apply a factorial ANOVA, the assumptions of parametric statistics should hold.

The factorial ANOVA produces an F-ratio for each factor as well as each interaction betweenfactors. In the case of a two-way factorial ANOVA that includes level and position as betweensubjects factors, there will be three F-ratio produced

An F-ratio for position

An F-ratio for half

An F-ratio for the interaction of position and half

The F-ratio represents the ratio of the variation of players of different groups to the variation of

players of the same group.
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The results of the factorial ANOVA firstly indicate whether each factor has a significantinfluence on the dependent variable. It is only necessary to use post-hoc tests for thosesignificant factors that are measured at more than two levels. Therefore, post-hoc testsshould be used to compare different positions but not to compare different halves. There areonly two halves. If the F-ratio for half is significant, the difference is obviously between thefirst half and the second half.

The results secondly indicate whether there is a significant interaction between factors. Aninteraction occurs where the pattern of distances covered by players of different positionsdiffers between players of different levels. This means that distance covered is influenced bythe combination of level and position.

With the mixed ANOVA, each type of result will be found in the tables entitled "Tests of withinsubjects effects" and "Tests of between subjects effects". Any interactions of between abdwithin subjects effects appear in the table of "Tests of within subjects effects".

An Example

Consider a group of 48 soccer players that are distinguished by position. There four positions(centre-back, wing-back, midfield and forward). The distance covered by the players ismeasured during both the first half and the second half. The mixed ANOVA can be used todetermine if the distance covered is significantly influenced by the half of the match beingplayed, position as well as the interaction between these two factors. If there is a significantdifference between positions, then post hoc tests can be employed to explore the significanceof differences between individual pairs of positions. If, however, half has a significantinfluence on distance covered, post-hoc tests are not required as this factor is only measuredat two levels.

Performing a Mixed ANOVA in SPSS

SPSS does not provide a specific facility for the mixed ANOVA but it can be achieved using arepeated measures ANOVA and introducing a between-subjects effect. From the Analysemenu select the General Linear Model sub-menu and from that select the RepeatedMeasures option. The GLM - Repeated Measures Define Factor(s) popup window willappear. Our datasheet will not contain a single variable representing a within-subjects factor.Instead, there will be a variable representing each level of the within subjects factor. We mustenter the factor name and number of levels into "Within Subject Factor Name" and "Numberof Levels" respectively. When we click on the Add button, this factor will be included but asyet it is undefined. When we click on Define, the GLM - Repeated Measures popup windowwill appear. We use this window to identify the variables representing each level of the withinsubjects factor. Once we transfer the variable representing the between-subjects factor into"Between-Subjects Factor(s)" the test will no longer be for a repeated measures ANOVA butfor a mixed ANOVA. The Post Hoc button allows the results of post hoc tests for between

subjects factor(s) to be included in the output. Note, however, that post hoc tests for within-subjects factor(s) are achieved through Options. The Options button provides a popupwindow that allows the user to have descriptive statisticsfor the different samples included inthe results. Other options allow Levene's test of homogeneity of variances to be done. Thereare various tables of output provided. The table entitled "Tests of Within Subjects Factors"shows the F-ratio and significance of within subjects factors as well as any interactionsinvolving within subjects factors (including interactions of between and within subjectsfactors). The table entitled "Tests of Between Subjects Effects" shows the F-ratio andsignificance of each between subjects factor as well as each interaction of two or morebetween-subjects factors. The degrees of freedom for each factor are important as are theerror degrees of freedom.

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The variable "level" has two values; 1 for elite and 2 for semi-professional.The variable "pos" has four values; 1 for centre-backs, 2 for wing-backs, 3 for midfielders and4 for forwards.The variable "distance" is the dependent variable.

SPSS does not provide a specific facility for the mixed ANOVA but it can be achieved using a

repeated measures ANOVA and introducing a between-subjects effect. From the Analysemenu select the General Linear Model sub-menu and from that select the RepeatedMeasures option. The GLM - Repeated Measures Define Factor(s) popup window willappear. We have one within-subjects factor to define and that is half. Half is measured attwo levels so we enter "Half" and 2 into "Within Subject Factor Name" and "Number of Levels"respectively. When we click on the Add button, this factor will be included but as yet it isundefined. When we click on Define, the GLM - Repeated Measures popup window willappear. We use this window to identify "first" and "second" as the two variables representingthe two levels of half. Once we transfer position into "Between-Subjects Factor(s)" the testwill no longer be for a repeated measures ANOVA but for a mixed ANOVA. The Post Hocbutton allows the results of post hoc tests for position to be included in the output. The tableentitled "Tests of Within Subjects Factors" shows the F-ratio and significance of half as wellas the interaction of half and position. The table entitled "Tests of Between Subjects Effects"

shows the F-ratio and significance of position. The results might be reported as follows.

A factorial ANOVA was applied to the data including half as a within subjects factor andposition as a between subjects factor. Players covered a significantly greater distance in thefirst half than in the second half (F1,44 = 23.7, P < 0.001). Position also had a significantinfluence on distance covered (F3,44 = 6.1, P < 0.001) with LSD post hoc tests revealing thatmidfielders covered a significantly greater distance than both centre-backs (P < 0.01) andforwards (P < 0.01). However, there was no significant interaction between position and half(F3,44 = 0.5, P > 0.05).



Exercise

Use a mixed ANOVA to determine the influence of half (first or second half) and level of play(elite or semi-professional) on the distance covered by soccer players.

Click here to access datasheet for this exercise (note this datasheet may already be open asit is the same data as for the previous example).

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