HFS3283 paired t tes-t and anova

HFS3283PAIRED T-TEST

&ONE WAY-ANOVA

DR. SHARIFAH WAJIHAH WAFA BTE SST WAFASchool of Nutrition and Dietetics

Faculty of Health [email protected]

KNOWLEDGE FOR THE BENEFIT OF HUMANITY

SCHOOL OF NUTRITION AND DIETETICS . FACULTY OF HEALTH SCIENCES

Topic Learning Outcomes

At the end of this lecture, the student should be able to:

1• Understand structure of research study appropriate

for ANOVA test

2• Understand how to evaluate the assumptions

underlying this test

3 • interpret SPSS outputs and report the results


Overviewt-tests

1. One-sample t-test2. Independent samples t-test3. Paired samples t-test

ANOVAs4. 1-way ANOVA5. 1-way repeated measures ANOVA6. Factorial ANOVA


Why a t-test or ANOVA?

•A t-test or ANOVA is used to determine whether a sample of scores are from the same population as another sample of scores.•These are inferential tools for examining differences between group means.• Is the difference between two sample means

‘real’ or due to chance?


t-tests

•One-sampleOne group of participants, compared with fixed, pre-existing value (e.g., population norms)•IndependentCompares mean scores on the same variable across different populations (groups)•PairedSame participants, with repeated measures


Major assumptions

•Normally distributed variables•Homogeneity of variance

In general, t-tests and ANOVAs are robust to violation of assumptions, particularly with large cell sizes, but don't be complacent.


Use of t in t-tests

•t reflects the ratio of between group variance to within group variance•Is the t large enough that it is unlikely that the two samples have come from the same population?•Decision: Is t larger than the critical value for t? (see t tables – depends on critical and N)


One-tail vs. two-tail tests

• Two-tailed test rejects null hypothesis if obtained t-value is extreme is either direction

• One-tailed test rejects null hypothesis if obtained t-value is extreme is one direction (you choose – too high or too low)

• One-tailed tests are twice as powerful as two-tailed, but they are only focused on identifying differences in one direction.


One sample t-test

• Compare one group (a sample) with a fixed, pre-existing value (e.g., population norms)

• Do uni students sleep less than the recommended amount?e.g., Given a sample of N = 190 uni students who sleep M = 7.5 hrs/day (SD = 1.5), does this differ significantly from 8 hours hrs/day ( = .05)?


One sample t-test


Independent groups t-test

• Compares mean scores on the same variable across different populations (groups)

• Do males & females differ in the amount of sleep they get?


Assumptions (Indep. samples t-test)

•LOM– IV is ordinal / categorical– DV is interval / ratio

•Homogeneity of Variance: If variances unequal (Levene’s test), adjustment made•Normality: t-tests robust to modest departures from normality, otherwise consider use of Mann-Whitney U test•Independence of observations (one participant’s score is not dependent on any other participant’s score)


Do males and females differ in in amount of sleep per night?


Independent samples t-test

•Comparison b/w means of 2 independent sample variables = t-test(e.g., what is the difference in Educational Satisfaction between male and female students?)

•Comparison b/w means of 3+ independent sample variables = 1-way ANOVA(e.g., what is the difference in Educational Satisfaction between students enrolled in four different faculties?)

PAIRED T-TEST



Paired samples t-test 1-way repeated measures ANOVA

• Same participants, with repeated measures• Data is sampled within subjects. Measures are

repeated e.g.,:

–Time e.g., pre- vs. post-intervention–Measures e.g., approval ratings of

brand X and brand Y• The paired t-test will show whether the differences

observed in the 2 measures will be found reliably in repeated samples.


Assumptions (Paired samples t-test)

• LOM:– IV: Two measures from same participants (w/in

subjects)• a variable measured on two occasions or• two different variables measured on the same occasion

– DV: Continuous (Interval or ratio)• Normal distribution of difference scores (robust to

violation with larger samples)• Independence of observations (one participant’s score is

not dependent on another’s score)


Example

• In this example, we want to compare the weight changes amongst obese children after 6 weeks went for weight management program.

• Five obese children are selected at random from the school A.

• We are interested in the following research question: Does an intervention have an effect on body weight of

the obese children? • The average weight, in both pre and post treatment

is recorded in columns 1 and 2 (see next slide) for each of 5 people (P1-P5):


19

X1 : 0 month

X2 : 6 month

P1: 52 51 4 1P2: 54.5 52 0.25 0P3: 53 52 1 0P4: 54.5 52 0.25 0P5: 56 53 4 1

54 52

1.9 0.4

1

211 )(

2

222 )(

n

sx2

2 )(

Example 4.1 (cont.) Column 1 Column 2 Column 3 Column 4


Example 4.1 (cont.)

• Unlike the independent samples t-test, on each row the numbers in columns 2 and 3 come from the same people.

• Person 2, for example, weigh of 64.5 kg in the pre-treatment, but lost to an average of 62 kg in the post-treatment (after all treatment was completed).

• It appears that this treatment may have improved body weight.


Example 4.1 (cont.)

• The paired t-test will allow us to see if the improvement that we see in this sample is reliable.

• If we selected another 5 obese childrenat random from the weight management program, would we still see an improvement?

• Without having to go through the trouble and expense of repeated sampling (called replication), we can estimate whether the difference in the 2 means is so large in magnitude that we would likely find the same result if we chose another 5 persons.


Example 4.1 (cont.)

12

212121

22

21

nSSrSS

txxxxxx

, df = n-1

Example 4.1 (cont.)

• This paired “t” needs a couple more values that we have not yet computed.

• First, we need to find the Standard Deviation of X1 and X2, called Sx1 and Sx2.

• These are simply the square-root of the variances ( and ). 6325.04.02

1xS 3784.19.12

2xS

Example 4.1 (cont.)

• Second, we need to find the correlation between the pre and post-treatment ( ), or likewise columns 2 and 3.

• Another section will illustrate how to compute a correlation.

• This computation is somewhat long, so we’ll avoid it for now.

• I’ll just tell you the correlation is: rx1x2=0.9177.

• Any scientific or statistical calculator can get you this answer.

21xxr


Example 4.1 (cont.)

15)3784.1)(6325)(.9177(.29.14.

5254t 4.78, df = 4

Example 4.1 (cont.)

• Finally, this computed “t” statistic must be compared with the critical value of the t-distribution.

• The critical value of the “t” is the highest magnitude we should expect to find if there is really no difference between the population means of X1 and X2, or in other words, no difference between weight in the pre and post treatment in the weight management program.

• Since we expect there should be a weight loss, this is a 1-tailed test.


Example 4.1 (cont.)

• The C.V. t(4), α=.05 = 2.132, therefore we reject the null hypothesis because the absolute value of our “t” at 4.78 is greater than the critical value.

• This is a 1-tailed t-test, so we must verify this conclusion by noting that the mean of the post treatment at 52kg, is lower than than the mean of the pre-treatment average of 54 kg.


Example 4.1 (cont.)

Our research conclusion states the facts in simple terms:

mean weight was decreased significantly from the pre-treatment

(M = 54) to the post-treatment (M = 52), t(4) = 4.78, p < .05 (one-tailed).


Example 4.1 Using SPSS

• First, we must setup the variables in SPSS.• Although not strictly necessary, it is good practice to

give a unique code to each participant (“personid”). • Unlike the independent samples t-test, the paired t-

test has separate entries for 2 dependent variables, rather than an independent and dependent:– DependentVariable1 = pretreat

(for Pre-treatment)– DependentVariable2 = posttreat

(for Post-treatment)




• In our example, the variables are setup as follows in the SPSS variable view:


• To run a Paired Samples t Test in SPSS, click Analyze > Compare Means > Paired-Samples T Test.




• You will be presented with the Paired-Samples T Test dialogue box, as shown below.

• transfer the variables pretreat and posttreat into the Paired Variables: box.


• Paired Sample Statistics Table– The first table, titled Paired Samples Statistics, is where SPSS Statistics

has generated descriptive statistics for the variables. You could use the results here to describe the characteristics of the pre- and post treatment.



• Paired Samples Test Table– The Paired Samples Test table is where the results of the dependent t-

test are presented.

• You are essentially conducting a one-sample t-test on the differences between the groups.


https://statistics.laerd.com/spss-tutorials/one-sample-t-test-using-spss-statistics.php


• You should focus your attention first of the mean values for the pre and the post treatment.

• As before, the means (Pre-treatment=54 and Post-treatment=52) give us our conclusion.

• Namely, we conclude that weight decreased from the pre to the post season.

• The statistics tell us that our conclusion is true not only for this sample of 5 persons, but also for other samples of 5 persons in the weight management program.


Example 4.1 Conclusion

• Our test is 1-tailed, so we must divide the 2-tailed probability provided by SPSS in half (p=.009/2 = .0045).

• When expressed to 2 significant digits, this value will round to “.00” and as a result the lowest value that can be represented in APA style is “p<.01.”

• In short, we can now write our conclusion as follows:Weight of obese children decreasedsignificantly from the pre-treatment

(M = 54) to the post-treatment (M = 52), t(4) = 4.78, p < .01 (one-tailed).


t-tests

• Difference between a set value and a variable → one-sample t-test

• Difference between two independent groups → independent samples t-test = BETWEEN-SUBJECTS

• Difference between two related measures (e.g., repeated over time or two related measures at one time) → paired samples t-test = WITHIN-SUBJECTS


ANOVA


Introduction to ANOVA(Analysis of Variance)

• Extension of a t-test to assess differences in the central tendency (M) of several groups or variables.

• DV variance is partitioned into between-group and within-group variance

• Levels of measurement:• Single DV: metric,• 1 or more IVs: categorical



Introduction

• ANOVA is an acronym for ANalysis Of VAriance. • The adjective oneway means that there is a single

variable that defines group membership (called a factor).

• Comparisons of means using more than one variable is possible with other kinds of ANOVA analysis.


When to use a One-way ANOVA • One-way ANOVA is a generalization of the

independent samples t-test.

• Recall that the independent samples t-test is used to compare the mean values of 2 different groups.

• A One-way ANOVA does the same thing, but it has the advantage of allowing comparisons between more than 2 groups.


When to use a Oneway ANOVA (continued)

• In health, for example, we often want to contrast several conditions in an experiment; such as a control, a standard treatment, and a newer “experimental” treatment.

• Because Oneway ANOVA is simply a generalization of the independent samples t-test, we use this procedure (to follow) to recalculate our previous 2 groups example.

• Later, we will do an example with more than 2 groups.


Example

• Let’s return to our example of the nasi lemak (NL) vs. roti canai (RC) diet BUT now we add up another one which in nasi dagang diet (ND)

• Our research question is:

“Is there any weight gain difference between a 1-week exclusive diet of either NL, RC or ND?”


Example -con’t Column 3 Column 4 Column 5

X1 : NL X2 : RC X3 : ND

1 3 3 1 1 02 4 2 0 0 12 4 3 0 0 02 4 3 0 0 03 5 4 1 1 12 4 3 2 2 2

0.4 0.4 0.4

211 )(

222 )(

1

n

sx2

2 )(

2 3

233 )(

H0: μ1= μ2= μ3Ha: At least one pair is different.


Comparing the groups

• Averages within groups: – NL: 2– RC: 4– ND: 3

• Total average: • Variance around the mean matters for comparison. • We must compare the variance within the groups to

the variance between the group means.

3555

)3(5)4(5)2(5


Variance within and between groups

• Sum of squares within groups: – SSW = = (1-2)2+(2-2) 2+(2-2) 2+(2-2) 2+……+(5-4) 2+…… = 6

• Compare it with sum of squares between groups: – SSB = – = (2-3) 2 +(2-3) 2+(2-3) 2+……+(4-3) 2+…… = 5 (2-3) 2 + 5 (4-3) 2 + (3-3) 2 = 10– Comparing these, we also need to take into account the

number of observations and sizes of groups

2)( jj

])(X[ 2Tj


Adjusting for group sizes

• Divide by the number of degrees of freedom

• F: • , reject H0 if this is large

MSGMSW

Both are estimates of population variance of error under H0

n: number of observationsK: number of groups

1SSGMSGK

MSBSSB

SSWMSWn K

MSWSSW

MSW

MSB


Example– Test statistic thresholds

• If populations are normal, with the same variance, then we can show that under the null hypothesis,

• Reject at confidence level if

1,~ K n KMSG FMSW

1, ,K n K

MSG FMSW

The F distribution, with K-1 and n-K degrees of freedom

Find this value in a table

MSB

MSW

MSB

MSW


Example– con’t

94.8 9.4813 3

SSWMSWn K

MSW6

15-30.5

52.43 26.21 3 1

SSGMSGK

MSBSSB 10

3-15.0

26.2 2.769.48

MSGMSW

MSB

MSW

5.0

0.510.0 F3-1,15-3,0.05 = 3.89

Thus we reject the null hypothesis in our case.


Example – ANOVA table

Next, we need to fill-in the so-called ANOVA table:

Source of variation

Sum of squares

Deg. of freedom

Mean squares

F ratio

Between groups

SSB K-1 MSB MSB/MSW

Within groups

SSW n-K MSW

Total SST n-1


Example – ANOVA table (cont.)

Next, we need to fill-in the so-called ANOVA table:

Source of

Variance(SV)

Sum of Squares

(SS)

Degrees of

Freedom(df)

Mean Squares

(MS)

F-ratio(F)

Critical Value(CV)

Reject Decision(Reject?)

Between 10 3-1= 2 10/2= 5.0 5.0/0.5= 10.0

3.89 Is F-ratio > CV ?

YES

Within 6 15-3= 12 6/12= 0.5

Total 16 15-1= 14



Example– F-tests

• In our case, when diet has no effect, differences between diet are entirely due to chance. Numerator and denominator will be similar. F-ratio should have value around 1.00

• When the diet does have an effect then the between-diet differences (numerator) should be larger than chance (denominator). F-ratio should be noticeably larger than 1.00


Example Using SPSS• 1. Quick Data Check

– We first want to get an idea of what our data basically look like. A nice option for the data at hand is a running a histogram of weight for each of the three groups separately. The screenshot below walks you through doing so.

1

2

3

4


Example Using SPSS

The shapes of the frequency distributions are normally distributed


Normality assumptionA

BC


Example 1 Using SPSS (cont.)

2. Running SPSS One-Way ANOVA

1

2

3

45



2. Running SPSS One-Way ANOVA (con’t.)Under button. Tick the checkbox as shown below:

4



2. Running SPSS One-Way ANOVA (con’t.)Click the button. Tick the Descriptive checkbox in the –Statistics– area, as shown below:

5


Example Using SPSS (cont.)

3. SPSS One-Way ANOVA Output– Two sections (boxes) appear in the output: Descriptives– “N” in the first column refers to the number of cases used for calculating the

descriptive statistics. These numbers being equal to our sample sizes tells us that there are no missing values on the dependent variable.

– The mean weights are the core of our output. After all, our main research question is whether these differ for different diets.



• 3. SPSS One-Way ANOVA Output-con’t– The second section, ANOVA table

• The significance level is 0.003 (p <0.01), and, therefore, there is a statistically significant difference in the mean weight gain between the different diets.

• which of the specific groups differed? • Find this out in the Multiple Comparisons table which contains the results

of post-hoc tests.


Example 1 Using SPSS (cont.)3. SPSS One-Way ANOVA Output-con’t• The table below, Multiple Comparisons, shows which groups differed from

each other. • there is a significant difference in weight gain between NL diet and RC diet

(p = 0.002). However, there were no differences between NL diet and ND diet (p=0.105), as well as between RC diet and ND diet (p=0.105)


In the Literature

• First and foremost, report descriptive statistics.

• Regarding the significance test,report– the F value;– df1, the numerator degrees of freedom;– df2, the denominator degrees of freedom;– the p value


In the Literature

• There was a statistically significant difference between groups as determined by one-way ANOVA (F(2,12) = 10.0, p<0.01). A Tukey post-hoc test revealed that weight gain was statistically significantly lower in NL diet (M= 2.00, SD= 0.71) compared to RC diet (M= 4.00, SD = 0.71, p <0.01). However, ND diet (M= 3.00, SD= 0.71) did not significantly differ from NL and RC diet.

Diet Mean (SD) t statistics (df) p-valueNasi Lemak 2.00 (0.71) 10.0(2,12) 0.003

Roti Canai 4.00 (0.71)

Nasi Dagang 3.00 (0.71)

Table 1: Type of diet associated with weight gain


Steps in solving One-Way ANOVA post hoc Test Problems - 1

The following is a guide to the decision process for answering homework problems about one-way ANOVA post hoc test problems:

Is the dependent variable ordinal or interval level and does independent variable define groups?

Incorrect application of

a statistic

No

Compute the skewness, and kurtosis for the variable to test assumption of normality.

Yes



The following is a guide to the decision process for answering homework problems about one-way ANOVA post hoc test problems:

Is the dependent variable ordinal or interval level and does independent variable define groups?


a statistic

No

Compute the skewness, and kurtosis for the variable to test assumption of normality.

Yes



Yes

No

Assumption of normality satisfied? (skew, kurtosis between -1.0 and + 1.0)

No

Sample size 10+ in each group to apply Central Limit Theorem?


a statisticYes

Compute the one-way ANOVA with Tukey HSD post hoc option selected



Is the p-value for the Tukey HSD post hoc test <= alpha?

Examine Tukey HSD post hoc test result

False Is the p-value for the F ratio test <= alpha?

No

Yes


recap

• FIVE assumptions– dependent variable should be measured at the interval or

ratio level (i.e., they are continuous). – independent variable should consist of two or more

categorical, independent groups.– should have independence of observations, which means

that there is no relationship between the observations in each group or between the groups themselves.

– homoscedasticity: the dependent variable has the same variance within each population;

– normality: the dependent variable is Gaussianly distributed within each population;


AnyQuestio

ns?

Concepts?

Data & Analytics

HFS3283 paired t tes-t and anova