04a independent sample t-test

8/6/2019 04a Independent Sample T-Test

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Independent Samples tIndependent Samples t--TestTest

(or 2(or 2--Sample tSample t--Test)Test)

Advanced Research Methods in PsychologyAdvanced Research Methods in Psychology

-- lecturelecture --

Matthew RockloffMatthew Rockloff


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When to use the independentWhen to use the independent

samples tsamples t--testtest The independent samples tThe independent samples t--test is probably thetest is probably the

single most widely used test in statistics.single most widely used test in statistics.

It is used toIt is used to compare differencescompare differences betweenbetweenseparate groups.separate groups.

In Psychology, these groups are often composedIn Psychology, these groups are often composedby randomly assigning research participants toby randomly assigning research participants toconditions.conditions.

However, this test can also be used to exploreHowever, this test can also be used to exploredifferences in naturally occurring groups.differences in naturally occurring groups.

For example, we may be interested in differencesFor example, we may be interested in differencesof emotional intelligence between males andof emotional intelligence between males andfemales.females.


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When to use the independent samplesWhen to use the independent samples

tt--testtest (cont.)(cont.)

Any differences between groups can beAny differences between groups can beexplored with the independent texplored with the independent t--test, astest, as

long as the tested members of eachlong as the tested members of eachgroup are reasonably representative ofgroup are reasonably representative ofthe populationthe population. [1]. [1]

[1][1]There are some technicalThere are some technicalrequirements as well. Principally,requirements as well. Principally,each variable must come from aeach variable must come from a

normal (ornearlynormal) distribution.normal (ornearlynormal) distribution.


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Example 3.1Example 3.1 (cont.)(cont.)

At the end of the week, we measureAt the end of the week, we measure

weight gain by each participant.weight gain by each participant.

Which diet causes more weight gain?Which diet causes more weight gain?

In other words, the null hypothesis is:In other words, the null hypothesis is:

Ho: wt. gain pizza diet =wt. gain beer diet.Ho: wt. gain pizza diet =wt. gain beer diet.


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Why?Why?

The null hypothesis is the opposite ofThe null hypothesis is the opposite of

what we hope to find.what we hope to find.

In this case, our research hypothesis isIn this case, our research hypothesis is

that there ARE differences between the 2that there ARE differences between the 2

diets.diets.

Therefore, our null hypothesis is thatTherefore, our null hypothesis is that

there are NO differences between these 2there are NO differences between these 2

diets.diets.


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XX11 : Pizza: Pizza XX22 : Beer: Beer

11 33 11 1122 44 00 00

22 44 00 00

22 44 00 00

33 55 11 11

22 44

0.40.4 0.40.4

!'1

2

11 )( ''

!' 2

2

22 )( ''

!''

!

n

sx

2

2)(


Column 3 Column 4


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The first step in calculating theThe first step in calculating the

independent samples tindependent samples t--test is to calculatetest is to calculate

thethe variancevariance andand meanmean in each condition.in each condition.

In the previous example, there are a totalIn the previous example, there are a total

of 10 people, with 5 in each condition.of 10 people, with 5 in each condition.

Since there are different people in eachSince there are different people in eachcondition, these samples arecondition, these samples are

independentindependent of one another; of one another;

giving rise to the name of the test.giving rise to the name of the test.


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The variances and means are calculatedThe variances and means are calculatedseparately for each conditionseparately for each condition

(Pizza and Beer).(Pizza and Beer). A streamlined calculation of theA streamlined calculation of the variancevariance forfor

each condition was illustrated previously.each condition was illustrated previously.(See Slide 7.)(See Slide 7.)

In short, we take each observed weight gainIn short, we take each observed weight gainfor the pizza condition, subtract it from thefor the pizza condition, subtract it from themean gain of the pizza dieters (mean gain of the pizza dieters ( 22) and) andsquare the result (seesquare the result (see column 3column 3).).

!'1


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Next, add up column 3 and divide by theNext, add up column 3 and divide by the

number of participants in that conditionnumber of participants in that condition

(n(n11 = 5) to get the= 5) to get the sample variancesample variance,,

The same calculations are repeated forThe same calculations are repeated for

the beer condition.the beer condition.

4.02!

xs


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FormulaFormula

The formula forThe formula for

thethe independent samples tindependent samples t--testtest is:is:

112

2

1

2

21

21

''

n

S

n

S

t

xx

, df = (n1-1) + (n2-1)


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Some theorySome theory

The t is calculated under theThe t is calculated under the

assumption, called theassumption, called the null hypothesisnull hypothesis,,

that there are no differences between thethat there are no differences between the

pizza and beer diet.pizza and beer diet.

If this were true, when we repeatedlyIf this were true, when we repeatedlysample 10 people from the populationsample 10 people from the population

and put them in our 2 diets, most oftenand put them in our 2 diets, most often

we would calculate a t of 0.we would calculate a t of 0.


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Some theorySome theory -- Why?Why?

Look again at the formula for the t.Look again at the formula for the t.

Most often the numerator (XMost often the numerator (X11--XX22) will be) will be

0, because the0, because the meanmean of the twoof the twoconditions should be theconditions should be the samesame under theunder the

null hypothesis.null hypothesis.

That is, weight gain is the same underThat is, weight gain is the same under

both the pizza and beer diet.both the pizza and beer diet.


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Some theorySome theory -- WhyWhy (cont.)(cont.)

Sometimes the weight gain might be a bitSometimes the weight gain might be a bithigher under the pizza diet, leading to ahigher under the pizza diet, leading to apositive t value.positive t value.

In other samples of 10 people, weightIn other samples of 10 people, weightgain might be a little higher under thegain might be a little higher under thebeer diet, leading to a negative t value.beer diet, leading to a negative t value.

TheThe important pointimportant point, however, is that, however, is thatunder the null hypothesis we shouldunder the null hypothesis we shouldexpect that most t values that weexpect that most t values that wecompute are close to 0.compute are close to 0.


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Some theorySome theory (cont.)(cont.)

Our computed tOur computed t--value is not 0, but it is in factvalue is not 0, but it is in fact

negative (t(8) =negative (t(8) = --4.47).4.47).

Although the tAlthough the t--value is negative, this should notvalue is negative, this should notbother us.bother us.

Remember that the tRemember that the t--value is onlyvalue is only -- 4.474.47

because we named the pizza diet Xbecause we named the pizza diet X11 and theand the

beer diet Xbeer diet X22..

This is, of course,This is, of course, completely arbitrarycompletely arbitrary..

If we had reversed our order of calculation, withIf we had reversed our order of calculation, with

the pizza diet as Xthe pizza diet as X22 and the beer diet as Xand the beer diet as X11, then, then

our calculated tour calculated t--value would be positive 4.47.value would be positive 4.47.


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Example 3.1Example 3.1 (again)(again) CalculationsCalculations

The calculated tThe calculated t--value is 4.47 (notice, Ivevalue is 4.47 (notice, Ive

eliminated the unnecessary eliminated the unnecessary -- sign sign),),

and the degrees of freedom are 8.and the degrees of freedom are 8. In the research question we did notIn the research question we did not

specify which diet should cause morespecify which diet should cause more

weight gain, therefore this tweight gain, therefore this t--test is a sotest is a so--

calledcalled 22--tailed t.tailed t.


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Example 3.1Example 3.1 (again)(again) CalculationsCalculations

In theIn the last steplast step, we need to find the, we need to find thecritical value for a 2critical value for a 2--tailed t with 8tailed t with 8

degrees of freedom.degrees of freedom. This is available from tables that are inThis is available from tables that are in

the back of any Statistics textbook.the back of any Statistics textbook.

Look in the back for Critical Values ofLook in the back for Critical Values of

the tthe t--distribution, or something similar.distribution, or something similar. The value you should find is:The value you should find is:

C.V.C.V. t(8), 2t(8), 2--tailedtailed = 2.31= 2.31..


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The calculated tThe calculated t--value of 4.47 is larger invalue of 4.47 is larger inmagnitude than the C.V. of 2.31, therefore wemagnitude than the C.V. of 2.31, therefore wecan reject the null hypothesis.can reject the null hypothesis.

Even for a results section of journal article, thisEven for a results section of journal article, thislanguage is a bit too formal and general. It islanguage is a bit too formal and general. It ismore important to state the research result,more important to state the research result,namely:namely:

Participants on the Beer diet (Participants on the Beer diet (MM= 4.00)= 4.00)gained significantly more weight thangained significantly more weight thanthose on the Pizza diet (those on the Pizza diet (MM= 2.00), t(8) == 2.00), t(8) =4.47, p < .05 (two4.47, p < .05 (two--tailed).tailed).


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Example 3.1Example 3.1 (concluding comment)(concluding comment)

Repeat from previous slide:Repeat from previous slide:

Participants on the Beer diet (Participants on the Beer diet (MM= 4.00)= 4.00)gained significantly more weight thangained significantly more weight thanthose on the Pizza diet (those on the Pizza diet (MM= 2.00), t(8) == 2.00), t(8) =4.47, p < .05 (two4.47, p < .05 (two--tailed).tailed).

Making this conclusion requiresMaking this conclusion requiresinspection of the mean scores forinspection of the mean scores foreach condition (Pizza and Beer).each condition (Pizza and Beer).


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Example 3.1 UsingSPSSExample 3.1 UsingSPSS

First,First, the variables must be setup in the SPSSthe variables must be setup in the SPSSdata editor.data editor.

We need to include both the independent andWe need to include both the independent and

dependent variables.dependent variables. Although it is not strictly necessary, it is goodAlthough it is not strictly necessary, it is good

practice to give each person a unique codepractice to give each person a unique code(e.g., personid):(e.g., personid):


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Example 3.1 UsingSPSSExample 3.1 UsingSPSS(cont.)(cont.)

In the previous example:In the previous example:

Dependent VariableDependent Variable

== wtgainwtgain (or weight gain)(or weight gain)

Independent VariableIndependent Variable == dietdiet

Why?Why? The independent variable (diet)The independent variable (diet) causescauses

changeschanges in the dependent variablein the dependent variable

(weight gain).(weight gain).


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Next,Next, we need to provide codes thatwe need to provide codes thatdistinguish between the 2 types of diets.distinguish between the 2 types of diets.

By clicking in the grey box of the L

abelBy clicking in the grey box of the L

abelfield in the row containing the dietfield in the row containing the dietvariable, we get a popvariable, we get a pop--up dialog thatup dialog thatallows us toallows us to codecode the diet variable.the diet variable.

Arbitrarily, the pizza diet is coded as dietArbitrarily, the pizza diet is coded as diet1 and the beer diet is diet 2.1 and the beer diet is diet 2.

Any other 2 codes would work, but theseAny other 2 codes would work, but thesesufficesuffice

See next slide.

See next slide.


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Example 3.1 UsingSPSSExample 3.1 UsingSPSS(coding)(coding)


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Example 3.1 UsingSPSSExample 3.1 UsingSPSS(data view)(data view)

Moving to the dataMoving to the dataview tab of the SPSSview tab of the SPSSeditor, the data iseditor, the data is

entered.entered. Each participant isEach participant is

entered on a separateentered on a separateline; a code is enteredline; a code is entered

for the diet they werefor the diet they wereon (1 = Pizza, 2 =on (1 = Pizza, 2 =Beer); and the weightBeer); and the weightgain of each isgain of each is

entered, as followsentered, as follows


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for the diet they werefor the diet they wereon (1 = Pizza, 2 =on (1 = Pizza, 2 =Beer); and the weightBeer); and the weightgain of each isgain of each is

entered, as followsentered, as follows


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for the diet they werefor the diet they wereon (1 = Pizza, 2 =on (1 = Pizza, 2 =Beer); and the weightBeer); and the weightgain of each isgain of each isentered, as followsentered, as follows


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(command syntax)(command syntax)

Next,Next, thethe command syntaxcommand syntax for an independentfor an independent

tt--test must be entered into the command editor.test must be entered into the command editor.

The format for the command is as follows:The format for the command is as follows:tt--test groupstest groups IndependentVariableIndependentVariable((Level1Level1,,Level2Level2))

variables=variables=DependentVariableDependentVariable..

You must substitute the names of theYou must substitute the names of the

independent and dependent variables, as wellindependent and dependent variables, as well

as the codes for the 2 levels of the independentas the codes for the 2 levels of the independent

variable. In our example, thevariable. In our example, the syntaxsyntax should beshould be

as per the next slideas per the next slide


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(command syntax) (cont.)(command syntax) (cont.)

After running this syntax, the following

output appears in the SPSS output viewer

See next slide.


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Example 3.1: SPSSOutput viewerExample 3.1: SPSSOutput viewer

Independent Samples Test


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SPSS gives theSPSS gives the meansmeans for each of thefor each of theconditions (Pizza Mean = 2 and Beerconditions (Pizza Mean = 2 and BeerMean = 4).Mean = 4).

In addition, SPSS provides aIn addition, SPSS provides a tt--valuevalue ofof --4.47 with 8 degrees of freedom.4.47 with 8 degrees of freedom.

These are the same figures that wereThese are the same figures that werecomputed by hand previously.computed by hand previously.

However, SPSSHowever, SPSS does notdoes not provide aprovide acritical value.critical value.

Instead, anInstead, an exact probabilityexact probability is providedis provided(p = .002).(p = .002).


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As long as this pAs long as this p--value falls below thevalue falls below the

standard of .05, we can declare astandard of .05, we can declare a

significant difference between our meansignificant difference between our meanvalues.values.

Since .002 is below .05 we can conclude:Since .002 is below .05 we can conclude:

Participants on the Beer diet (Participants on the Beer diet (MM= 4.00)= 4.00)gained significantly more weight thangained significantly more weight than

those on the Pizza diet (those on the Pizza diet (MM= 2.00),= 2.00),

t(8) = 4.47,t(8) = 4.47, p < .01p < .01 (two(two--tailed).tailed).


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The SPSS output also displaysThe SPSS output also displays Levenes TestLevenes TestforforEquality of VariancesEquality of Variances (see the first 2(see the first 2columns in second table on slide 30).columns in second table on slide 30).

Why?Why? Strictly speaking, the tStrictly speaking, the t--test is only valid if wetest is only valid if we

havehave approximately equal variancesapproximately equal variances within eachwithin each

of our two groups.of our two groups. In our example, this was not a problem becauseIn our example, this was not a problem because

the 2 variances were exactly equal (Variancethe 2 variances were exactly equal (VariancePizza = 0.04 and Variance Beer = 0.04).Pizza = 0.04 and Variance Beer = 0.04).


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However, if this test is significant,However, if this test is significant,

meaning that the pmeaning that the p--value given is lessvalue given is less

than .05, then we should choose thethan .05, then we should choose thebottom line when interpreting ourbottom line when interpreting our

results.results.

This bottom line makes slightThis bottom line makes slight

adjustments to the tadjustments to the t--test to accounttest to accountfor problems when there are notfor problems when there are not

equal variances in both conditions.equal variances in both conditions.


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The practical importance of thisThe practical importance of this

distinction is small.distinction is small.

Even if variances are not equal betweenEven if variances are not equal betweenconditions, the hand calculations weconditions, the hand calculations we

have shown will most often lead to thehave shown will most often lead to the

correct conclusion anyway, and use ofcorrect conclusion anyway, and use of

the top line is almost alwaysthe top line is almost alwaysappropriate.appropriate.


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Independent Samples tIndependent Samples t--TestTest

(or 2(or 2--Sample tSample t--Test)Test)

Advanced Research Methods in PsychologyAdvanced Research Methods in Psychology

-- Week 2 lectureWeek 2 lecture --

Matthew RockloffMatthew Rockloff

04a independent sample t-test

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