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Independent t-tests. Let’s think…. Comparing 2 means drawn from the same population If we sampled enough, what would we expect the mean difference to be? What would influence the accuracy of this expectation?. Comparing 2 means.

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let s think
Let’s think…
  • Comparing 2 means drawn from the same population
    • If we sampled enough, what would we expect the mean difference to be?
    • What would influence the accuracy of this expectation?
comparing 2 means
Comparing 2 means
  • Estimating the mean of the sampling distribution of differences in the means
  • Estimating the standard error of the differences in the means
    • Getting a little complicated…
comparing 2 means4
Comparing 2 means
  • Estimating the standard error of the differences in the means
    • Must assume equal population variance in the two samples
      • This assumption of independent t-tests must be tested
    • Then…

the SD of the distribution of differences between 2 sample means

comparing 2 means5
Comparing 2 means
  • T-test for 2 independent samples

Difference between sample means

SEm of difference between sample means

Note:

comparing means of two groups or pairs of data
Comparing means of two groups or pairs of data
  • Our single sample example: Our 8th grade % body fat compared to the national norm.
    • National Norm: 23%
    • Our class: 20%

Is the observed difference between groups a

reflection of treatment effect or only

random sampling error?

recall
Recall
  • Larger sample size ==>
  • Less variability in population ==>

reduced variability in the

distribution of sampling means

extending to a comparison of two sample means
Extending to a comparison of two sample means
  • With larger samples, the less likely that the observed difference in sample means is attributable to random sampling error
extending to a comparison of two sample means9
Extending to a comparison of two sample means
  • With larger samples, the less likely that the observed difference in sample means is attributable to random sampling error
  • With reduced variability among the cases in each sample, the less likely that the observed difference in sample means is attributable to random sampling error
extending to a comparison of two sample means10
Extending to a comparison of two sample means
  • With larger samples, the less likely that the observed difference in sample means is attributable to random sampling error
  • With reduced variability among the cases in each sample, the less likely that the observed difference in sample means is attributable to random sampling error
  • Larger the observed difference between two sample means, the less likely that the observed difference in sample means is attributable to random sampling error
types of data sets
Independent

separate groups

males & females

experienced & unexperienced

injury & non-injured

fit & unfit

young & old

Uncorrelated data sets

Unmatched groups

Types of Data Sets
types of data sets12
Independent

separate groups

males & females

experienced & unexperienced

injury & non-injured

fit & unfit

young & old

Uncorrelated data sets

Unmatched groups

Dependent

repeated measures on the same individual

test - retest

pre - post

time 1 - time 2

Correlated data pairs

Matched groups (pairs)

Types of Data Sets
independent t test uncorrelated t test
Independent t-test(Uncorrelated t-test)
  • Apply when there is no reason to assume correlation between the cases in the two groups
  • Question: Does CHO supplementation increase aerobic endurance?
    • IV: dietary CHO level
    • DV: time to exhaustion on bicycle ergometer
steps to independent t test
Steps to independent t-test
  • Set  (0.05)
  • Set sample size
    • Two randomly selected groups of subjects
      • n = 10 in each group
    • Group 1: Regular diet
    • Group 2: CHO supplemented diet
  • Set Ho (null hypothesis)
set statistical hypotheses
Ho

Null hypothesis

Any observed difference between the two groups will be attributable to random sampling error.

HA

Alternative hypothesis

If Ho is rejected, the difference is not attributable to random sampling error

perhaps diet???

Set statistical hypotheses
steps to independent t test16
Steps to independent t-test
  • Set  (0.05)
  • Set sample size ( n = 10 per group)
  • Set Ho
  • Test all subjects with same protocol (bicycle ergometer)
steps to independent t test18
Steps to independent t-test
  • Set  (0.05)
  • Set sample size ( n = 10 per group)
  • Set Ho
  • Test all subjects with same protocol (bike)
  • Calculate descriptive statistics for each group
steps to independent t test20
Steps to independent t-test
  • Set  (0.05)
  • Set sample size ( n = 10 per group)
  • Set Ho
  • Test all subjects with same protocol (bike)
  • Calculate descriptive statistics for each group
  • Compare the two group means
slide21

From WMK Trochim

http://trochim.human.cornell.edu/kb/stat_t.htm

  • Note mean difference = in all three comparisons
  • Need to evaluate mean difference according to variability in sets of scores.
how to compare the groups
How to compare the groups
  • Even if the two groups were the same (drawn from the same population or no treatment effect), do not expect the two means to be the same.
  • Need a measure of expected variability against which the mean difference could be compared.
recall estimating standard error of the mean from sample statistics
Recall: Estimating Standard Error of the Mean from sample statistics

There is a SEM relative to each sample.

estimated standard error of the difference between 2 independent means25
Estimated Standard Error of the Difference between 2 independent means

Estimate of the expected variability in

when samples are of size n (mean of these differences = ???

for the diet study
For the diet study
  • SDx = 3.542 (reg diet)
  • SDy = 2.860 (CHO diet)
  • n for both groups = 10
  • Calculate SEm for each of the groups
    • Reg diet: 1.12
    • CHO diet: 0.90
  • Calculate Sdm in this situation
    • 1.44
t test for independent samples28
t-test for independent samples

Calculate for the diet study data

t test for independent samples29
Xm = 38.9

Ym = 44.2

Sdm = 1.44

Mean diff

-5.3

tobs

-3.68

t-test for independent samples
evaluating t observed with the t distribution
Evaluating tobserved with thet distribution
  • Recall: a distribution of t values is not normally distributed
  • Follows a t distribution
    • concept of degrees of freedom (df)
    • comparing two independent means
      • df = (nx -1) + (ny -1)
evaluating t observed with the t distribution31
Evaluating tobserved with thet distribution
  • A distribution of t values is not normally distributed
  • Follows a t distribution
    • concept of degrees of freedom (df)
    • comparing two independent means
      • df = (nx -1) + (ny -1)
      • becomes df = 2 ( n - 1) if groups are equal in n
slide32

Standard deviation  1

  • Leptokurtic (narrower peak, larger tails than z-dist)
  • shape depends on df
slide33

tcritical: the value of t that must

be exceeded to classify a

difference between means

as statistically significant.

t critical depends on df and one vs two tailed test
tcritical depends on df, , and one vs two tailed test
  • http://duke.usask.ca/~rbaker/Tables.html

For our diet study:

df = 2 (10 - 1) = 18

 = 0.05

two-tailed test

tcrit = ???

t critical depends on df and one vs two tailed test36
tcritical depends on df, , and one vs two tailed test
  • http://duke.usask.ca/~rbaker/Tables.html

For our diet study:

df = 2 (10 - 1) = 18

 = 0.05

two-tailed test

tcrit =  2.101 Note the 

concept of evaluating t obs vs t crit
Concept of evaluatingtobs vs tcrit

df = 18

 = 0.05

tcrit

Area = 0.025

Area = 0.025

-2.101

2.101

concept of evaluating t obs vs t crit38
Concept of evaluatingtobs vs tcrit

df = 18

 = 0.05

Region

of

Rejection

Region

of

Rejection

Region

of

Non-rejection

-2.101

2.101

concept of evaluating t obs vs t crit39
Concept of evaluatingtobs vs tcrit

df = 18

 = 0.05

tcrit

Area = 0.025

Area = 0.025

X

tobs = -3.68

-2.101

2.101

decision
Decision
  • Since tobs = -3.68 is beyond the tcrit value of -2.101, our decision is to ...

Falls in the region of rejection

decision41
Decision
  • Since tobs = -3.68 is beyond the tcrit value of -2.101, our decision is to reject Ho and accept HA that ...
decision42
Decision
  • Since tobs = -3.68 is beyond the tcrit value of -2.101, our decision is to reject Ho and accept HA that
    • the differences between the groups in time to exhaustion reflects differences in
      • diet??
      • Poor methodological control?
      • Some other confounding variable?
reporting t test in table
Reporting t-test in table

Table 1. Descriptive statistics of time to exhaustion

( in minutes) for the two diets.

*

Note: * indicates significant difference, p 0.05

reporting t test graphically
Reporting t-test graphically

Figure 1. Mean time to exhaustion with different diets.

reporting t test graphically46
Reporting t-test graphically

Figure 1. Mean time to exhaustion with different diets.

reporting t test in text
Reporting t-test in text

Descriptive statistics for the time to exhaustion for the

two diet groups are presented in Table 1 and

graphically in Figure 1. A t-test for independent

samples indicated that the 44.2 ( 2.9) minute time to

exhaustion for the CHO group was significantly

longer than the 38.9 ( 3.5) minutes for the regular

diet group (t18 = - 3.68, p 0.05). This represents

a approximate 10% increase in time to exhaustion with

the CHO supplementation diet.

In discussion, address whether the statistically

significant difference is clinically relevant

interpretation example
Interpretation Example

People with family care physician as primary physician had less chance of dying than those with specialist as primary physician.

Recruiting lecture to SIU School of Medicine

borg rpe scale
Borg RPE Scale

Subjects asked

to rate “intensity”

of physical

activity, by

selecting a number

corresponding to

perceived

exertion.

meaningful vs significant difference
Meaningful vs significantdifference

The caffeine group

reported lower RPE’s

than the placebo group.

These findings imply

that caffeine gives the

subject some kind of

euphoric stimulus

while exercising.

type i type ii errors
Type I &Type IIerrors

False negatives and

False positives

summary of the independent t test
Summary of theindependent t-test
  • Utilize when the assumption of no correlation between the groups is valid
  • Compares the difference in means by evaluating the observed magnitude of the mean difference to the expected variability in the magnitude of the mean differences when Ho is true.