Chapter 4 Introduction to Hypothesis Testing Introduction to Hypothesis Testing.
Hypothesis Testing
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Transcript of Hypothesis Testing
Hypothesis Testing
• Start with a question:Does the amount of credit card debt differ between households in rural areas compared to households in urban areas?
Population 1All Rural Households
Population 2All Urban Households
Null Hypothesis: H :
Alternate Hypothesis: HA : 1≠2
Collect Data to Test Hypothesis
Population 1All Rural Households
Population 2All Urban Households
Are the sample means consistent with H0?
Take Random Sample(n1)
1x
Take Random Sample(n2)
2x
Summary Data
Summary Rural Summary Urban
Difference in means = $735
How likely is it to get a difference of $735 or greater if Ho is true? This probability is called the p-value.
If small then reject Ho.
3412
6299
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1
s
x
2467
7034
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s
x
P-Value
The probability of observing a difference between sample means as or more extreme as that observed if the null hypothesis is true.
When this probability is small we declare that the two population means are significantly different.
P< 0.05 is conventional cutoff
Note: P-value and significance level are the same
Computing P-Value for Testing Differences Between 2 Means
test statistic:
21
21
11nn
s
xxt
p
Sp is pooled standard deviation, a weighted average of SD for each group
Under Ho t follows a t-distribution with n1+ n2 -2 degrees of freedom (DF)
Point estimator for
Variability in point estimate
Observations
21
21
11nn
s
xxt
p
• If Ho is true then t-values should center around 0
• A large difference between sample means will lead to a large t-value
• A small standard error will lead to a large t-value
• results from large sample sizes (n1 and n2)
• results from small variation in the population
Assumptions for T-Test
1. Each of 2 populations follow a normal distribution
2. Data sampled independently from each population
– Example of lack of independenceMeasure visual acuity in left and right eye
3. The population variances are the same for each population.
The t-test is “robust” to violation of assumptions 1 and 3.
Robust – the assumptions do not need to hold exactly
* SAS CODE FOR CREDIT CARD EXAMPLE;
DATA credit;INFILE DATALINES;INPUT balance live @@;DATALINES;9619 1 5364 1 8348 1 7348 1 381 1 2998 1 1686 1 1962 1 4920 1 5047 16644 1 7644 1 11169 1 7979 1 3258 1 8660 1 7511 1 14442 1 4447 1 6550 17581 2 12545 2 7959 2 2563 2 6787 25071 2 9536 2 4459 2 8047 2 8083 22153 2 8003 2 6795 2 5915 2 7164 29980 2 8718 2 8452 2 4935 2 5938 2;
Used when inputing more than one obs per line
PROC MEANS DATA=credit ; CLASS live; VAR balance;
The MEANS Procedure
Analysis Variable : balance
Nlive Obs N Mean Std Dev Minimum Maximum
1 20 20 6298.85 3412.31 381.0000000 14442.00
2 20 20 7034.20 2467.36 2153.00 12545.00
PROC TTEST DATA=credit ; CLASS live; VAR balance; OUTPUT The TTEST Procedure
Statistics
Lower CL Upper CL Lower CLVariable live N Mean Mean Mean Std Dev Std Dev
balance 1 20 4701.8 6298.9 7895.9 2595 3412.3balance 2 20 5879.4 7034.2 8189 1876.4 2467.4balance Diff (1-2) -2641 -735.3 1170.8 2433.4 2977.6
Means for each group and the difference
PROC TTEST DATA=credit ; CLASS live; VAR balance;
OUTPUTT-Tests
Variable Method Variances DF t Value Pr > |t|
balance Pooled Equal 38 -0.78 0.4397
balance Satterthwaite Unequal 34.6 -0.78 0.4401
T-statistic and P-value
DF = n1+n2 – 2
Conclusion: Means are not significantly different (p=.44)
PROC TTEST DATA=credit ; CLASS live; VAR balance;
OUTPUT
Tests if variances are different between groups
Equality of Variances
Variable Method Num DF Den DF F Value Pr > F
balance Folded F 19 19 1.91 0.1666
Your Turn
• Page 256 of Le
• Compares cotinine levels from 8 infants from parents who smoke and 7 infants from parents who do not smoke.
• What are the 2 populations?
• Write down in words and symbols the null and alternate hypothesis
• Write and run the SAS code to perform the t-test
• Compare the SAS output with the calculations on page 256
• What is the p-value for the test?
Matched Pair Data
• Each subject serves as own control• Half of patients start out on treatment 1, other half on
treatment 2• Outcome is measured at end of first period• Patients are switched to other treatment (usually after
a “washout” period).• Outcome is measured at end of second period• Analyses is based on within subject differences
Matched Pair Data Examples
• Data on twins• Pre-post tests• Data on pairs of eyes, left versus right foot, etc
Matched Pair Data
• Analyses reduced to a 1-sample problem• Differences are computed for each pair
– di = outcome when on treatment 1 minus outcome when on treatment 2
ns
dt
1
Large values indicate differences in treatments
Matched Pair Example
• Question: Does intake of oat bran lower your cholesterol?
• LDL cholesterol measured on 14 subjects
– After period on cornflake diet
– After period on oat bran diet
• Data on page 273 of Le
DATA oatbran; INFILE DATALINES; INPUT subject $ cornflakes oatbran ; oatcorndif = oatbran - cornflakes;DATALINES; 1 4.61 3.84 2 6.42 5.57 3 5.40 5.85 4 4.54 4.80 5 3.98 3.68 6 3.82 2.96 7 5.01 4.41 8 4.34 3.72 9 3.80 3.49 10 4.56 3.84 11 5.35 5.26 12 3.89 3.73 13 2.25 1.84 14 4.24 4.14 ;
*Running Matched Pair T-test using proc means: ;
PROC MEANS DATA=oatbran N MEAN STDERR T PRT ;
VAR oatcorndif
OUTPUTThe MEANS Procedure
Variable N Mean Std Error t Value Pr > |t|
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
cornflakes 14 4.4435714 0.2589319 17.16 <.0001
oatbran 14 4.0807143 0.2824898 14.45 <.0001
oatcorndif 14 -0.3628571 0.1084984 -3.34 0.0053
Tvalue = mean/se
Conclusion: Oat bran significantly reduces cholesterol (p<.01)
*Running Matched Pair T-test using PROC TTEST;PROC TTEST; VAR oatcorndif;RUN;
No class variable so performing one sample t-test. Tests if mean is 0.
Match Pair Data- Your TurnFemale killdeer lay four eggs each spring. A scientist claims that the egg that hatches first yields a larger bird than the one that hatches last. To test his claim, he weighs the oldest and youngest of eight families with the following results:
Family Oldest Youngest
1 2.92 2.90
2 3.58 3.68
3 3.39 3.33
4 3.29 3.06
5 3.44 3.30
6 3.13 2.99
7 3.22 3.26
8 3.80 3.51
Test the researcher’s hypothesis using the data above? What is the null and alternative hypothesis? What is the p-value for the test?
Issues with hypothesis testing
• Significance does not imply causality– Need a proper prospective experiment
• Significance does not imply practical importance– Trivial but significant differences
• Run lots of tests, will find significant difference by chance– With α = 0.05, expect 1 in 20 results to be sig. by chance
Issues with hypothesis testing
• Large p-values because sample size is small– Effect could exist but we may not have a large enough
sample size
• Outliers may cause problems
Issues With Hypothesis Testing
What is the population of inference?
Example: A statistics class of n=15 women and n=5 men yield the following exam scores:
Women: mean = 90% SD = 10%Men: mean = 85% SD = 11%
Test the hypothesis that women did better on the exam then men.
Hypothesis tests and Confidence Intervals
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Two sampletest statistic:
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CI for differencein means: