Psychology 203

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Psychology 203

• Semester 1, 2007
• Week 9
• Lecture 14

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Related Samples t-test
Extra Bonus Material Confidence intervals aka,
the gentle art of estimation
Gravetter Wallnau, Chapter 11
and Chapter 12
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What are related samples?

• All the people in the sample are related, right?

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How did they feel ?
June 26, 2006, Germany, World Cup Soccer,
Australia v Italy
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Then this happened
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How do they feel now?
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Repeated Measures

• Two sets of data are collected from the same
  individuals
• Same variable measured twice, at different times,
  in same individuals
• Note more complicated repeated measures designs
  include more than two measurements

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Why repeated measures rock

• Same individuals in all conditions
• No risk of getting two random samples who differ
  a lot on a relevant variable
• e.g. IQ differences btwn groups in a memory expt
• Participants in both samples are perfectly
  matched (because they are the same people!)
• Faking it

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Matched Subjects

• Approximating repeated measures
• One-to-one match of participants in both groups
  on variable of interest
• e.g. comparing two kinds of ways of teaching
  stats
• Match participants for prior maths courses
  completed
• Maths grades, IQ
• Repeated measures perfect matching

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Other names

• Two sets of scores, with each score in one sample
  directly related to a score in the other
• Related samples
• Correlated samples
• Paired samples
• Repeated measures most common form

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t-test for related samples

• Just like single sample t-test!

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Hypotheses for related-samples t-test

• The null hypothesis?
• The result of the game has no effect on mood
• So the mean of the difference scores should be?
• H0 ?D 0
• Some people feel a bit more miserable after the
  game, some feel a bit happier
• it all balances out to produce no overall
  difference

Zero
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Hypotheses for related-samples t-test

• The alternate/experimental hypothesis?
• The result of the game has an effect on mood
• H1 ?D ? 0
• People will feel consistently more miserable
  after the game (or consistently happier)
• Set ?.05, 2-tailed

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t-statistic for related-samples
t-statistic for single sample
sample mean of the difference scores
population mean of the difference scores
estimated standard error for the difference scores
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t-test for related samples

• Our data

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Calculating the standard error

• Just the same as for single-samples t-test
• Calculate variance/standard deviation of
  difference scores

or
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Calculating the standard error

• Calculate estimated standard error for difference
  scores

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Calculating t
Compare our t to the critical value of t for
df4, ?.05, 2-tailed
Our effect is not significant and we fail to
reject the null hypothesis
tobtained lt tcrit
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Calculating effect size for related-samples t-test
large effect
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Paired-samples t-test in SPSS
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Reporting the results

• Losing did not significantly change the mean mood
  ratings of Australian soccer fans, M 4.6,
  SD3.78, t(4)2.72, p.053, d1.21.
• The mood ratings of Australian soccer fans were
  lower after losing the game, M3.8, SD3.11, than
  they were before, M8.4, SD1.14. However, the
  effect was only marginally significant,
  t(4)2.72, p.053, d1.21.

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One-tailed test

• Aussie soccer fans will feel more miserable after
  losing the game

Compare our t of 2.72 to the critical value of t
for df4, ?.05, 1-tailed
Our effect is significant and we reject the null
hypothesis
tobtained gt tcrit
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Assumptions of related-samples t-test

• Observations within each treatment must be
  independent
• Population distribution of difference scores must
  be normal

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Choosing between a repeated an independent
measures design

• The advantages of repeated measures
• Fewer participants, often more efficient
• Can study changes over time
• Reduces problems cause by individual differences

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Repeated measures controls for individual
differences
Repeated measures
MD5
SS8
t(2)4.35, p lt .05
n3
large individual differences in mood
Independent measures
MBefore26
SSBefore114
n3
MAfter21
SSAfter86
n3
t(4)0.87, p gt .05
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Choosing between a repeated an independent
measures design

• The advantages of repeated measures
• Fewer participants, often more efficient
• Can study changes over time
• Reduces problems cause by individual differences
• The main disadvantage
• Factors other than treatment causing changes over
  time

i.e. a more sensitive, powerful design
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Dealing with other factors

• Other factors include
• Fatigue
• Learning
• Time (e.g. pre 9/11 versus post 9/11)
• Repetition
• Counterbalancing
• e.g. half participants do one condition first and
  the other condition second vice versa
• so effects other than treatment are spread
  equally across both conditions

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Estimation
221 000

• www.oztam.com.au/html/index.htm

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Error, precision and confidence

• The difference between the sample the
  population is sampling error
• The greater the sampling error the less accurate
  your estimate will be
• Two types of estimates-
• Point estimate
• Interval estimate

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Betting at the races!

• Two types of bets-
• To win
• To place (each-way)

Chances of winning anything small
Point estimate
Payout big
Chances of winning anything better
Payout less
Interval estimate
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Estimate Types

• Point estimate estimating a single number e.g.
  Dockers will be 4 on AFL Ladder
• precise estimate
• but cant be too confident about it
• Interval estimate estimating a range of values
  e.g. Dockers will be in the Top 8
• not as precise
• can be more confident youll be right

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Estimation Hypothesis testing

• Both inferential procedures that use samples to
  answer questions about populations
• Answer slightly different questions
• Hypothesis testing Is there an effect?
• Estimation How much effect is there?

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When to use estimation

• After a hypothesis test has already rejected the
  null hypothesis
• e.g. clinical significance, does drug reduce
  cholesterol to safe levels
• Already know there is an effect but want to know
  the size
• Want some basic information about a population
• e.g. which tv shows they are watching

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Using the t-statistic for estimation

• Hypothesis testing
• Given our sample data and our estimate of error,
  how likely is it that our difference is due to
  chance i.e. null hypothesis is true
• Estimation
• Given our sample data and estimates of error,
  what do we estimate the population parameter
  (e.g. mean) to be?

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Using the t-statistic for estimation
set this using null hypothesis
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sample parameter (mean)
population parameter (mean)
t
estimated standard error
but now this is the value we want to estimate
If we could set this to be some value then we
could work out population mean
sample parameter (mean)
population parameter (mean)
estimated standard error)

(t x
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A reasonable value for t

• Use the known distributions of t-values to set a
  reasonable or highly probable value of t
• What is the most probable value of t?

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Remember our t-shirt study?

• Do people prefer t-shirts worn by models showing
  a genuine smile?
• Yes, mean proportion was 62. Significantly
  greater than chance 50 (single sample t-test).
• Now we can ask, how often do we estimate people
  in general (the population) will choose a t-shirt
  associated with a genuine smile?
• Our data- M0.62, sM.022, n10

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Point estimate of the population mean using the
t-statistic
Our best estimate of t is 0
The sample mean is always our best point estimate
of the population mean
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Confidence

• How confident are you that the population mean
  really is exactly 62?
• Not that confident!
• We would be more confident if we could say it
  falls in a range of values, e.g. between 58 and
  66
• i.e. an interval estimate
• How confident are you it is in this range?

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Confidence Intervals

• Determine confidence needed as a probability e.g.
  85
• Then use t to define a range of values within
  which our population mean is estimated to fall
• Instead of using the most common t (i.e. 0) we
  now select a value of t that matches our required
  level of confidence

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Calculating a confidence interval

• Decide on 80 confidence
• What are the values of t that define a region
  with 80 under the curve?
• df n-1 10-1 9

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Calculating a confidence interval
Use the value(s) of t defined by your confidence
level
Confidence interval is created with the sample
mean in the middle
80 Confidence Interval
Population mean is between 59 and 65
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Interpreting the confidence interval

• If we did our experiment again would we get the
  same confidence interval?
• No, because our Mean Standard error would be
  slightly different
• What if we got a freakishly extreme sample?
• 80 of our samples will produce CIs that contain
  ?

Sample 1
Sample 2
CI Interval 1
CI Interval 2
CI includes ?
CI does not include ?
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Confidence Intervals in SPSS
Socceroos Mood Data
Bigger sample produces a narrower interval i.e.
greater precision
Lowering confidence level produces narrower
interval
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Uses of Confidence Intervals

• Estimates of effect size, in terms of the scale
  you used to measure the effect
• e.g. T-shirts associated with genuine smiles were
  preferred between 59 and 65 (9-15 increase
  over chance)
• easier to understand interpret than Cohens D
  etc.
• Can use for hypothesis testing
• e.g. If the CI of difference scores includes 0,
  this suggests you do not have a significant
  difference
• e.g. 95 CI for the difference in the mood before
  and after the game was -0.096 to 9.29.