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Unit 2: Comparing Two GroupsIn Unit 1, we learned the basic
process of statistical inference using tests and confidence
intervals. We did all this by focusing on a single proportion.In
Unit 2, we will take these ideas and extend them to comparing two
groups. We will compare two proportions, two independent means, and
paired data.Chapter 5:Comparing Two Proportions5.1: Descriptive
(Two-Way Tables)5.2: Inference with Simulation-Based Methods5.3:
Inference with Theory-Based MethodsPositive and Negative
PerceptionsConsider these two questions:Are you having a good
year?Are you having a bad year?
Do people answer each question in such a way that would
indicated the same answer? (e.g. Yes for the first one and No for
the second.)Researchers questioned 30 students (randomly giving
them one of the two questions).They then recorded if a positive or
negative response was given.Is this an observational study or
randomized experiment?
Positive and Negative PerceptionsObservational unitsThe 30
students VariablesQuestion wording (good year or bad
year)Perception of their year (positive or negative)Which is the
explanatory and which is the response?Positive and Negative
PerceptionsIndividualType of QuestionResponseIndividualType of
QuestionResponse1Good YearPositive16Good YearPositive2Good
YearNegative17Bad YearPositive3Bad YearPositive18Good
YearPositive4Good YearPositive19Good YearPositive5Good
YearNegative20Good YearPositive6Bad YearPositive21Bad
YearNegative7Good YearPositive22Good YearPositive8Good
YearPositive23Bad YearNegative9Good YearPositive24Good
YearPositive10Bad YearNegative25Bad YearNegative11Good
YearNegative26Good YearPositive12Bad YearNegative27Bad
YearNegative13Good YearPositive28Good YearPositive14Bad
YearNegative29Bad YearPositive15Good YearPositive30Bad
YearNegativeRaw Data in a SpreadsheetA two-way table organizes data
Summarizes two categorical variables Also called contingency table
Are students more likely to give a positive response if they were
given the good year question?
Two-Way TablesGood YearBad YearTotalPositive
response15419Negative response3811Total181230Conditional
proportions will help us better determine if there is an
association between the question asked and the type of response.We
can see that those given the positive question were more likely to
respond positively.Two-Way TablesGood YearBad YearTotalPositive
response15/18 0.834/12 0.33 19Negative response3811Total181230We
can use segmented bar graphs to see this association.Remember that
variables are associated if the conditional proportion of the
outcomes for one group differ from the conditional proportion of
outcomes in other groups.
Segmented Bar Graphs
Those responding to the good year question were more likely to
answer positively (83% to 33%) than those responding positively to
the bad year question. The statistic we will be using to measure
this is the difference in proportions.0.83 - 0.33 = 0.50 higher for
the good year question than the bad year question.
PerceptionsIn the next section we will conduct tests of
significance to compare two proportions and I want to give you a
preview of that here. We will assume there is no association
between the variables (i.e. the two population proportions are the
same) and decide if two sample proportions differ enough to
conclude this would be very unlikely just by random chance.
A Sneak Peak at Comparing Two Proportions: Simulation-Based
ApproachHypothesesNull Hypothesis: There is no association between
which question is asked and the type of response. (The proportion
of positive responses will be the same in each group. ) Alternative
Hypothesis: There is an association between which question is asked
and the type of response. (The proportion of positive responses
will be different in each group. )
ResultsGood YearBad YearTotalPositive response15 (83%)4
(33%)19Negative response3811Total181230The difference in
proportions of positive responses is 0.83 0.33 = 0.50.How likely is
a difference this great or greater if the type of question asked
made no difference in how the student would respond?Random
ReassignmentNotice that 19 students gave a positive response. If
the null hypothesis is true, these 19 would have given a positive
response no matter which question was asked.Therefore, under a true
null hypothesis, we can randomly place these 19 people into either
group and they will still give a positive response. This replicates
the random assignment that was done in the experiment. We will also
keep constant the 18 that receive the positive question and 12 that
receive the negative question.Good YearBad YearTotalPositive
responserandomrandom19Negative responserandomrandom11Total181230You
can think about this random reassignment with the raw data as well.
It doesnt matter which question was asked, the responses will be
the same. Therefore, we can shuffle the type of question and leave
the responses fixed. This is equivalent to keeping the same column
and row totals and just shuffling the inside of the two-way table
as described earlier.IndividualType of
QuestionResponseIndividualType of QuestionResponse1Good
YearPositive16Good YearPositive2Good YearNegative17Bad
YearPositive3Bad YearPositive18Good YearPositive4Good
YearPositive19Good YearPositive5Good YearNegative20Good
YearPositive6Bad YearPositive21Bad YearNegative7Good
YearPositive22Good YearPositive8Good YearPositive23Bad
YearNegative9Good YearPositive24Good YearPositive10Bad
YearNegative25Bad YearNegative11Good YearNegative26Good
YearPositive12Bad YearNegative27Bad YearNegative13Good
YearPositive28Good YearPositive14Bad YearNegative29Bad
YearPositive15Good YearPositive30Bad YearNegativeRandom
ReassignmentI did this once and found a difference in the
proportions of positive responses for the two questions of 0.50
0.83 = 0.33Good YearBad YearTotalPositive response9 (50%)10
(83%)19Negative response9211Total181230Random ReassignmentI did
this again and found a difference in the proportions of positive
responses for the two questions of 0.61 0.67 = 0.06Good YearBad
YearTotalPositive response11 (61%)8 (67%)19Negative
response7411Total181230Random ReassignmentI did this again and
found a difference in the proportions of positive responses for the
two questions of 0.67 0.58 = 0.09Good YearBad YearTotalPositive
response12 (67%)7 (58%)19Negative response6511Total181230Random
ReassignmentIn my three randomizations, I have yet to see a
difference in proportions that is as far away from zero as the
observed difference of 0.5.Lets do some more randomizations to
develop a null distribution.
Random ReassignmentAfter 1000 randomizations, only 7 were as far
away from zero as our observed proportion.
ConclusionSince we have a p-value of 7/1000 or 0.007, we can
conclude the alternative hypothesis and say we have strong evidence
that how the question is phrased affects the response.21AppletsLets
look at how this is done in two applets Simulation for Two
ProportionsSimulation for Multiple Proportions
Exploration 5.1: Murderous Nurse?
Example 5.2: Swimming With Dolphins
Is swimming with dolphins therapeutic for patients suffering
from clinical depression?Researchers recruited 30 subjects aged
18-65 with a clinical diagnosis of mild to moderate depression.
Discontinued antidepressants and psychotherapy 4 weeks prior to and
throughout the experiment30 subjects went to an island near
HondurasRandomly assigned to two treatment groupsSwimming with
DolphinsBoth groups engaged in one hour of swimming and snorkeling
each day. One group swam in the presence of dolphins and the other
group did not.Participants in both groups had identical conditions
except for the dolphinsAfter 2 weeks, each subjects level of
depression was evaluated, as it had been at the beginning of the
study The response variable is if the subject achieved substantial
reduction in depression.
Swimming with DolphinsObservational units The 30 subjects with
mild to moderate depression.Explanatory variableSwimming with
dolphins or notResponse variableReduction in depression or notAre
the variables quantitative or categorical? Swimming with DolphinsIs
this study an observational study or an experiment? Are the
subjects in this study a random sample from a larger
population?Swimming with DolphinsResultsSwimming with
DolphinsDolphingroupControl groupTotalImproved10 (67%)3 (20%)13Did
Not Improve51217Total151530
The difference in proportions of improvers is 0.67 0.20 =
0.47.There are two possible explanations for an observed difference
of 0.47.A tendency to be more likely to improve with dolphinsThe 13
subjects were going to show improvement with or without dolphins
and random chance assigned more improvers to the dolphinsSwimming
with DolphinsNull hypothesis: Dolphins dont helpSwimming with
dolphins is not associated with substantial improvement in
depressionAlternative hypothesis: Dolphins helpSwimming with
dolphins increases the probability of substantial improvement in
depression symptoms
Swimming with DolphinsSwimming with DolphinsNull Hypothesis: The
probability someone exhibits substantial improvement after swimming
with dolphins is the same as the probability someone exhibits
substantial improvement after swimming without dolphins.Alternative
Hypothesis: The probability someone exhibits substantial
improvement after swimming with dolphins is higher than the
probability someone exhibits substantial improvement after swimming
without dolphins.
Swimming with DolphinsSwimming with DolphinsIf the null
hypothesis is true (dolphin therapy is not better) we would have 13
improvers and 17 non-improvers regardless of the group they were
in. Any differences we see between groups arise solely from the
randomness in the assignment to the groups.
Swimming with DolphinsWe can perform this simulation with cards.
13 black cards represent the improvers 17 red cards represent the
non-improversWe assume these outcomes would happen no matter which
treatment group subjects were in. Shuffle the cards and put 15 in
one pile (dolphin therapy) and 15 in another (control group)An
improver is equally likely to be assigned to each groupSwimming
with DolphinsIn the actual study, there were 10 improvers (diff of
0.47) in the dolphin group.We conducted 3 simulations and got 8, 5,
and 6 improvers in the dolphin therapy group. (notice the diff in
proportions)
Swimming with Dolphins
We did 1000 repetitions to develop a null distribution. Why is
it centered at about 0?What does each dot represent?Swimming with
Dolphins
Swimming with DolphinsA 95% confidence interval for the
difference in the probability using the standard deviation from the
null distribution is 0.467 + 2(0.178) = 0.467 + 0.356 or (0.111to
0.823)We are 95% confident that when allowed to swim with dolphins,
the probability of improving is between 0.111 and 0.823 higher than
when no dolphins are present. How does this interval back up our
conclusion from the test of significance?
Swimming with DolphinsCan we say that the presence of dolphins
caused this improvement? Since this was a randomized experiment,
and assuming everything was identical between the groups, we have
strong evidence that dolphins were the cause Can we generalize to a
larger population?Maybe mild to moderately depressed 18-65 year old
patients willing to volunteer for this studyWe have no evidence
that random selection was used to find the 30 subjects.
Swimming with DolphinsExploration 5.2: Contagious
Yawns?MythBusters investigated this. 50 subjects were ushered into
a small room by co-host Kari. She yawned as she ushered 34 in the
room and for 16 she didnt yawn. We will assume she randomly decided
who would received the yawns.
Comparing Two Proportions: Theory-Based ApproachSection
5.3Introduction
Just as with a single proportion, we can often predict results
of a simulation using a theory-based approach. The theory-based
approach also gives a simpler way to generate a confidence
intervals.
Smoking and Birth Gender
Smoking and GenderHow does parents behavior affect the sex of
their children?Fukuda et al., 2002 (Japan) found the following: 255
of 565 births (45.1%) where both parents smoked more than a pack a
day were boys. 1975 of 3602 births (54.8%) where both parents did
not smoke were boys.Other studies have shown a reduced male to
female birth ratio where high concentrations of other environmental
chemicals are present (e.g. industrial pollution, pesticides)
Smoking and GenderA segmented bar graph and 2-way tableLets
compare the proportions to see if the difference is statistically
significantly.
Smoking and GenderSmoking and GenderWhat are the observational
units in the study?What are the variables in this study?Which
variable should be considered the explanatory variable and which
the response variable? Can you draw cause-and-effect conclusions
for this study? Smoking and GenderOK to shuffle?In the last section
we re-randomized subjects to treatment groups to simulate the null
distribution.In this study the parents werent randomized to the
treatment, since its observational, but we can still represent the
null hypothesis of no association through randomization. Smoking
and GenderUse the 3S Strategy to asses the strength1. Statistic:
The proportion of boys born to nonsmokers minus boys born to
smokers is 0.548 0.451 = 0.097.Smoking and Gender2. Simulate: Use
the Multiple Proportions applet to simulate Many repetitions of
shuffling the 2230 boys and 1937 girls to the 565 smoking and 3602
nonsmoking parentsCalculate the difference in proportions of boys
between the groups for each repetition. Shuffling simulates the
null hypothesis of no associationSmoking and Gender3. Strength of
evidence: Nothing as extreme as our observed statistic ( 0.097 or
0.097) occurred in 5000 repetitions, How many SDs is 0.097 above
the mean?
Smoking and GenderNotice the null distribution is centered at
zero and is bell-shaped. This, along with its standard deviation
can be predicted using normal distributions.
Smoking and GenderWe can use either the Multiple Proportion
applet or the Theory-Based Inference applet to find the p-value
Smoking and GenderSmoking and GenderFrom our test of
significance, do we expect 0 to be in the interval of plausible
values for the difference in the population proportions?
Smoking and GenderAgain, either applet can be used to determine
a confidence interval.
We are 95% confident that the probability of a boy baby is 0.053
to 0.141 higher for families where neither parent smokes compared
to families with two smoking parents
Smoking and GenderWe can also write the confidence interval in
the form: statistic margin of error. Our statistic is the observed
sample difference in proportions, 0.097. We can find the margin of
error by subtracting the statistic (center) from the upper endpoint
or 0.141 0.097 = 0.044. 0.097 0.044 Is the margin of error about
the standard deviation?Smoking and GenderHow would the interval
change if the confidence level was 99%?
Smoking and GenderWritten as the statistic margin of error 0.097
0.058. Margin of error 0.058 for the 99% confidence interval0.044
for the 95% confidence interval
Smoking and GenderSmoking and Gender (0.141, 0.053) or 0.097
0.044 instead of (0.053, 0.141) or 0.097 0.044
The negative signs indicate the probability of a boy born to
smoking parents is lower than that for nonsmoking parents.
Smoking and GenderValidity Conditions of Theory-Based Same as
with a single proportion.Should have at least 10 observations in
each of the cells of the 2 x 2 table.Smoking ParentsNon-smoking
ParentsTotalMale25519752230Female31016271937Total56536024167Smoking
and GenderThe strong significant result in this study yielded quite
a bit of press when it came outSoon other studies came out which
found no relationship between smoking and genderOne article argued
that confounding variables likesocial factors, diet, environmental
exposure or stress were the reason for different studys results.
(These are all possible since it was an observational study.)
FormulasFormulasStrength of EvidenceAs the proportions move
farther away from each other, the strength of evidence increases.As
sample size increases, the strength of evidence increases.
Lets run this previous test using both the Simulation-Based and
the Theory-Based Applets.Donating BloodExploration 5.3Questions
1-14 (skip 2 and 5)