Comparisons between two means:

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Lecture 7

• Comparisons between two means
• Research questions about two separate or
  independent groups
• Research questions about two dependent or
  correlated groups

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Univariate vs. Multivariate

• Univariate analysis usually refers to one
  predictor variable and one outcome variable
• Is gender a predictor of pneumonia?
• Multivariate analysis usually refers to more than
  one predictor variable or more than one outcome
  variable being evaluated simultaneously.
• After adjusting for age, is gender a predictor of
  pneumonia?

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Difference vs. Association

• Some tests are designed to assess whether there
  are statistically significant differences between
  groups.
• Is there a statistically significant difference
  between the age of patients with and without
  pneumonia?
• Some tests are designed to assess whether there
  are statistically significant associations
  between variables.
• Is the age of the patient associated with the
  number of days in the hospital?

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Unmatched vs. Matched

• Some statistical tests are designed to assess
  groups that are unmatched or independent.
• Is the admission systolic blood pressure
  different between men and women?
• Some statistical tests are designed to assess
  groups that are matched or data that are paired.
• Is the systolic blood pressure different between
  admission and discharge?

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Hypothesis testing (Review)

• In any hypothesis testing situation, we first
  need to define the null hypothesis. It is under
  the null hypothesis that we will figure out how
  our test statistic is distributed.
• Knowing how our test statistic is distributed
  will allow us to use the corresponding
  probability distribution
• We start by assuming that null hypothesis is
  true. This gives us the acceptance and rejection
  regions defined at our chosen significance level
  for the distribution "under the null hypothesis"
  

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Hypothesis testing (Review)

• Note The null hypothesis, H0 and the alternative
  hypothesis Ha should be mutually exclusive and
  exhaustive.
• 2 types of errors.
• 1. Type I error reject a true null hypothesis.
  (commonly called a)
• 2. Type II error fail to reject a false null.
  (commonly called ß)
• Consider a jury's hypothesis
• H0 The defendant is innocent.
• Ha The defendant is guilty.
• Therefore
• Type I error would entail a false conviction of
  an innocent person.
• Type II error would entail letting a guilty
  person go free.

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Hypothesis Testing (Review)

• Significance tests are used to accept or reject
  the null hypothesis.
• This is done by studying the sampling
  distribution for a statistic.
• If the probability of observing your result is lt
  .05, reject the null
• If the probability of observing your result is gt
  .05, accept the null.
• There are many kinds of significance tests for
  different kinds of statistics. Today were going
  to discuss t-tests.

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2-Sample T-Tests

• Independent t-test
• Dependent t-test
• Picking the correct test

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t-test example

• We are interested in whether caffeine consumption
  improves peoples happiness.
• We randomly assign 25 people to drink decaf and
  25 people to drink regular coffee.
• Subsequently we measure how happy people are.
• Note The independent variable is categorical
  (youre in one group or the other), and there are
  only two groups.
• The dependent variable is continuouswe measure
  how happy people are on a continuous metric.

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t-test example (cont)

• Lets say we find that the control group has a
  mean score of 3 (SD 1) and the experimental
  group has a mean score of 3.2 (SD .9).
• Thus, there is a .20 difference between the two
  groups. 3.2 3.0 .2
• Two possibilities
• The .2 difference between groups is due to
  sampling error, not a real effect of caffeine. In
  other words, the two samples are drawn from
  populations with identical means and variances.
• The .2 difference between groups is due to the
  effect of caffeine, not sampling error. In other
  words, the two samples are drawn from populations
  with different means (and maybe different
  variances).

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Population for control group
Population for experimental group
These two populations have identical means and
variances
These two samples may or may not have identical
means and variances because of sampling error
hence, one sample mean might be .2 points higher
than the other
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t-test example (cont)

• We need to know how likely it is that we would
  observe a difference of .20 or higher if the null
  hypothesis is true.
• How can we do this?
• We can construct a sampling distribution of mean
  differencesassuming the null hypothesis is true.
• We can use this distribution to determine how
  large of mean difference we will observe on
  average when the population mean difference is
  zero.

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Assumptions 2-Sample T-Test

• Data in each group follow a normal distribution.
• For pooled test, the variances for each group are
  equal.
• The samples are independent. That is, who is in
  the second sample doesnt depend on who is in the
  first sample (and vice versa).

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Indep t-test formula
(For our purposes, always zero)
Actual difference observed.

• Standard Error of the Difference (between the
  means)
• difference expected between sample means
• how much we expect the sample means to differ
  purely by chance

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Ind. t-test Example
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Hypothesis Testing Steps (Ind. t)

• 1. Comparing xbar1 and xbar2, µ and s unknown.
• 2. H0 µ1 µ2 0 HA µ1 µ2 ? 0
• a .05, df n1n22 5 5 - 2 8
• tcritical 2.306
• 4. tcalculated -1.947
• 5. Accept (Fail to reject) the H0 .
• The research hypothesis was not supported.
• The weight of women in sororities (xbar111) does
  not differ significantly from that of other women
  (xbar127), t(8) -1.947, n.s..

(not needed if using SPSS)
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Ind. t-test Example (SPSS)
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Steps of Hypothesis Testing

• In an ideal research setting, we define a
  strategy to follow this order
• 1. Formulate hypothesis
• 2. Figure out what test statistic will test this
  hypothesis
• 3. Collect data
• 4. Perform the statistical test
• 5. Accept or reject the hypothesis.
• 4 elements in any hypothesis test.
• 1. A null hypothesis H0
• 2. An alternative hypothesis, Ha
• 3. A test statistic (how calculate, its
  distribution)
• 4. A rejection region (you want to be in the
  rejection region!)

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Caution on hypothesis testing

• In the jury, the social choice is as to what
  constitutes an acceptable risk -- to decide the
  probability of type I error.
• Generally, we refer to alpha (a) as the
  significance level, which is the highest limit we
  set on the probability of Type I error.
• In most statistical situations, by convention we
  set a 0.05. But in the criminal case, society
  may chose a 0.001 or even a 0.0001 such that
  we increase the probability of letting the guilty
  go free so as not to falsely convict an innocent
  person.
• If one reduces Type I error, one by necessity
  increases Type II error.

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What happens if samples arent independent?

• That is, they are
• dependent or correlated?

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Ways Pairing Can Occur

• When subjects in one group are matched with a
  similar subject in the second group.
• When subjects serve as their own control by
  receiving both of two different treatments.
• When, in before and after studies, the same
  subjects are measured twice.

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What is the effect of alcohol on useful
consciousness?

• Ten male subjects taken to a simulated altitude
  of 25,000 ft and given tasks to perform.
• For each, time (in seconds) at which useful
  consciousness ended was recorded.
• 3 days later, experiment was repeated one hour
  after subjects ingested 0.5 cm3 of 100-proof
  whiskey per pound of body weight.

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What is the effect of alcohol on useful
consciousness?
H0 ?D 0 vs. H0 ?D gt 0
Paired T for NoAlcohol - Alcohol
N Mean StDev SE Mean NoAlcohol
10 546.6 238.8 75.5 Alcohol
10 351.0 210.9 66.7 Difference
10 195.6 230.5 72.9 95 CI for
mean difference (30.7, 360.5) T-Test of mean
difference 0 (vs gt 0) T-Value 2.68
P-Value 0.013
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What is the effect of time on memory recall?

• 8 people were given 10 minutes to memorize a list
  of 20 nonsense words.
• Each was asked to list as many words as he or she
  could remember after 1 hour and again after 24
  hours.

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What is the effect of time on memory recall?
Paired T for 1hour - 24hour N
Mean StDev SE Mean 1hour 8
12.75 3.69 1.31 24hour 8
9.13 3.52 1.25 Difference 8
3.625 2.066 0.730 95 CI for mean
difference (1.897, 5.353) T-Test of mean
difference 0 (vs not gt 0) T-Value 4.96
P-Value
0.001
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Do males earn higher average starting salaries
than females?
(in 1,000s) Males Females 22
20 29 28 80 78 35
32 Sample Average 41.5 39.5
Real question is whether males and females in the
same job earn different average salaries. Better
to compare the difference in salaries in pairs
of males and females.
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Paired Study
Salaries (in 1,000s) Job Males Females Differe
nceM-F Non-Profit 22 20 2.0 Education
29 28 1.0 Doctor 80
78 2.0 Scientist 35 32
3.0 Averages 41.5 39.5 2.0
P-value How likely is it that a paired sample
would have a difference as large as 2,000 if the
true difference were 0?
Problem reduces to a One-Sample T-test on
differences!!!!
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The Paired-T Test Statistic

• If
• there are n pairs
• and the differences are normally distributed

Then The test statistic, which follows a
t-distribution with n-1 degrees of freedom, gives
us our p-value
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The Paired-T Confidence Interval

• If
• there are n pairs
• and the differences are normally distributed

Then The confidence interval, with t following
t-distribution with n-1 d.f. estimates the actual
population difference
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Data analyzed as Paired T
Paired T for M - F N Mean
StDev SE Mean M 4 41.5
26.2 13.1 F 4 39.5
26.1 13.1 Difference 4 2.000
0.816 0.408 95 CI for mean difference
(0.701, 3.299) T-Test of mean difference 0 (vs
not 0) T-Value 4.90
P-Value 0.016
P 0.016. Reject null. Sufficient evidence to
conclude that average starting salaries differ
between males and females.
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Now, Data analyzed as 2-Sample T
Two sample T for M vs F N Mean StDev
SE Mean M 4 41.5 26.2 13 F 4
39.5 26.1 13 95 CI for µ M - µ
F ( -43, 47) T-Test µ M µ F (vs not ) T
0.11 P 0.92 DF 6
P 0.92. Do not reject null. Insufficient
evidence to conclude that average starting
salaries differ between males and females.
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What happened?

• P-value from two-sample t-test is just plain
  wrong. (Assumptions not met.)
• We removed or blocked out the extra variability
  in the data due to differences in jobs, thereby
  focusing directly on the differences in salaries.
• The paired t-test is more powerful because the
  paired design reduces the variability in the data.

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Example 3

• You have a very important psychological
  question  Is apple pie preferred over pecan
  pie?  You give 9 people a slice of apple pie and
  a slice of pecan pie.  Since you are such a
  skilled researcher you present the slices of pie
  in a counterbalanced order across subjects.  You
  measure the number of grams of apple and pecan
  pie that each person eats.

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• What test? Related Samples t-test
• Level of significance?
• 2. State IV, levels of IV, and DV?
• 3. One-tailed or two-tailed test?
• 4. Hypotheses?
• 5. Critical value?
• df 8, tcrit 1.860

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Example Cont.
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Steps 6 7
t obs - 1.095 t obs lt t crit, therefore we fail
to reject the null hypothesis. Apple pie is not
preferred over pecan pie.
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