1. Basics of Pharmaceutical Statistics
Fundamentals of Hypothesis Testing
PREPARED BY: -
Dr. PORAS PATEL
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2. What is biostatistics
Statistics is the science and art of collecting, summarizing, and analyzing data that are subject to random variation.
Biostatistics is the application of statistics and mathematical methods to the design and analysis of health, biomedical, and biological studies.
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3. Different Tests of Significance
One-Sample (t-test or z-test)
Compares one sample mean versus a population mean
Two-Sample (t-test)
Compares one sample mean versus another sample mean
Independent t-tests (equal samples)
Dependent t-tests (dependent/paired samples)
Analysis of Variance (ANOVA)
Comparing several samples mean. It can be classified as One Way & Two Way and others like Ancova.
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4. How to properly use Biostatistics
Develop an underlying question of interest
Generate a hypothesis
Design a study (Protocol)
Collect Data
Analyze Data
Descriptive statistics
Statistical Inference
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5. Relationship between population and sample (Simple random sampling)
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6. Sampling Techniques Population Simple Random Sample Stratified Random
Systematic
Sampling
Cluster
Sampling
Convenience
Sampling
Bias free
sample
Bias free
sample
Biased
sample
Bias free
sample
Biased
sample
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7. Impact of Misuse of statistics
About 25% of biological research is flawed because of incorrect conclusions drawn from confounded experimental designs and misuse of statistical methods
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8. What is a Hypothesis?
A hypothesis is a claim (assumption) about the population parameter
Difference between the value of sample statistic and the corresponding hypothesized parameter value is called hypothesis testing.
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9. Hypothesis Testing Process
Assume the
population
mean age is 50.
Identify the Population
( )
Take a Sample
No, not likely!
REJECT
Null Hypothesis
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10. ... Therefore, we reject the null hypothesis that m = 50.
Reason for Rejecting H0
Sampling Distribution of
It is unlikely that we would get a sample mean of this value ...
... Therefore, we reject the null hypothesis that m = 50.
... if in fact this were the population mean. m
= 50 20
If H0 is true
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11. Components of Biostatistics
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12. Normal Distribution
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13. Normal distribution BELL-SHAPED
symmetrical about the mean (No skewness)
total area under curve = 1
approximately 68% of distribution is within one standard deviation of the mean
approximately 95% of distribution is within two standard deviations of the mean
approximately 99.7% of distribution is within 3 standard deviations of the mean
Mean = Median = Mode
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14. Empirical Rule
68%
About 68% of the area lies within 1 standard deviation of the mean
About 95% of the area lies within 2 standard deviations
This rule has been discussed
earlier. Emphasize that there is still 0.3% of the distribution falling outside the 3 standard deviation limits.
About 99.7% of the area lies within 3 standard deviations of the mean
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15. Copyright © Dr. Poras Patel
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16. Level of Significance, Is designated by , (level of significance)
Typical values are .01, .05, .10
Is selected by the researcher at the beginning
Provides the critical value(s) of the test
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17. The z-Test for Comparing Population Means
Critical values for standard normal distribution
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18. Level of Significance and the Rejection Region
I claim that mean CVD in the INDIA is atleast 3 out of 100 people! a
H0: m ³ 3 H1: m < 3
Critical Value(s)
Rejection Regions a
H0: m £ 3 H1: m > 3
a/2
H0: m = 3 H1: m ¹ 3
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19. Hypothesis Testing State the research question.
State the statistical hypothesis.
Set decision rule.
Calculate the test statistic.
Decide if result is significant.
Interpret result as it relates to your research question.
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20. Rejection & Non-Rejection Regions=
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21. The Null Hypothesis, H0 States the assumption (numerical) to be tested
e.g.: The average number of CVD in INDIA is at least three ( )
Is always about a population parameter ( ), not about a sample statistic ( )
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22. The Null Hypothesis, H0
(continued)
Begins with the assumption that the null hypothesis is true
Similar to the notion of innocent until proven guilty
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23. The Alternative Hypothesis, H1
Is the opposite of the null hypothesis
e.g.: The average number of CVD in INDIA is less than 3 ( )
Never contains the “=” sign
May or may not be accepted
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24. General Steps in Hypothesis Testing
e.g.: Test the assumption that the true mean number of of CVD in INDIA is at least three ( Known)
State the H0
State the H1
Choose
Choose n
Choose Test
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25. General Steps in Hypothesis Testing
(continued)
Set up critical value(s)
7. Collect data
8. Compute test statistic and p-value
9. Make statistical decision
10. Express conclusion
100 persons surveyed
Computed test stat =-2, p-value = .0228
Reject null hypothesis
The true mean number of CVD is less than 3 in human population.
Reject H0 a Z
-1.645
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26. The z-Test for Comparing Population Means
Critical values for standard normal distribution
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27. p-Value Approach to Testing
Convert Sample Statistic (e.g. ) to Test Statistic (e.g. Z, t or F –statistic)
Obtain the p-value from a table or computer
Compare the p-value with
If p-value , do not reject H0
If p-value , reject H0
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28. Comparison of Critical-Value & P-Value Approaches
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29. Result Probabilities H0: Innocent Jury Trial Hypothesis Test The Truth
Verdict
Innocent
Guilty
Decision H
True H
False
Do Not
Type II
Innocent
Correct
Error
Reject
1 - a
Error ( b ) H
Type I
Reject
Power
Guilty
Error
Correct
Error H
(1 - b ) ( a )
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30. Type I & II Errors Have an Inverse Relationship
If you reduce the probability of one error, the other one increases so that everything else is unchanged. b a
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31. Critical Values Approach to Testing
Convert sample statistic (e.g.: ) to test statistic (e.g.: Z, t or F –statistic)
Obtain critical value(s) for a specified from a table or computer
If the test statistic falls in the critical region, reject H0
Otherwise do not reject H0
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32. One-tail Z Test for Mean ( Known)
Assumptions
Population is normally distributed
If not normal, requires large samples
Null hypothesis has or sign only
Z test statistic
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33. Rejection Region Reject H0 Reject H0 Z Z H0: m ³ m0 H1: m < m0Z
Small values of Z don’t contradict H0 Don’t Reject H0 !
Z Must Be Significantly Below 0 to reject H0
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34. Example: One Tail Test H0: m £ 368 H1: m > 368
Q. Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showed = The company has specified s to be 15 grams. Test at the a = level.
368 gm.
H0: m £ H1: m > 368
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35. Finding Critical Value: One Tail
Standardized Cumulative Normal Distribution Table (Portion)
What is Z given a = 0.05?
.05 Z
.04
.06
1.6
.9495
.9505
.9515
.95
a = .05
1.7
.9591
.9599
.9608
1.8
.9671
.9678
.9686
1.645 Z
Critical Value = 1.645
1.9
.9738
.9744
.9750
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36. Example Solution: One Tail Test
H0: m £ H1: m > 368
Test Statistic:
Decision:
Conclusion:
a = 0.5
n = 25
Critical Value: 1.645
Reject
Do Not Reject at a = .05
.05
No evidence that true mean is more than 368
1.645 Z
1.50
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37. p -Value Solution p-Value is P(Z ³ 1.50) = 0.0668 Z 1.50
Use the alternative hypothesis to find the direction of the rejection region.
P-Value =.0668 Z
1.50
From Z Table: Lookup 1.50 to Obtain .9332
Z Value of Sample Statistic
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38. (p-Value = 0.0668) ³ (a = 0.05) Do Not Reject.
p -Value Solution
(continued)
(p-Value = ) ³ (a = 0.05) Do Not Reject.
p Value =
Reject
a = 0.05 Z
1.645
1.50
Test Statistic 1.50 is in the Do Not Reject Region
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39. Example: Two-Tail Test
Q. Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes showed = The company has specified s to be 15 grams. Test at the a = level.
368 gm.
H0: m = H1: m ¹ 368
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40. Example Solution: Two-Tail Test
H0: m = H1: m ¹ 368
Test Statistic:
Decision:
Conclusion:
a = 0.05
n = 25
Critical Value: ±1.96
Reject
Do Not Reject at a = .05
.025
.025
No Evidence that True Mean is Not 368
-1.96
1.96 Z
1.50
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41. (p Value = 0.1336) ³ (a = 0.05) Do Not Reject.
p-Value Solution
(p Value = ) ³ (a = 0.05) Do Not Reject.
p Value = 2 x
Reject
Reject
a = 0.05 Z
1.50
1.96
Test Statistic 1.50 is in the Do Not Reject Region
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42. Connection to Confidence Intervals
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43. What is a t Test?
Commonly Used Definition: Comparing two means to see if they are significantly different from each other
Technical Definition: Any statistical test that uses the t family of distributions t
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44. Independent Samples t Test
Use this test when you want to compare the means of two independent samples on a given variable
“Independent” means that the members of one sample do not include, and are not matched with, members of the other sample
Example:
Compare the average height of 50 randomly selected men to that of 50 randomly selected women
Independent Mean #1
Independent Mean #2
Compare using t test
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45. Dependent Samples t Test
Used to compare the means of a single sample or of two matched or paired samples
Example:
If a group of students took a math test in March and that same group of students took the same math test two months later in May, we could compare their average scores on the two test dates using a dependent samples t test
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46. Comparing the Two t Tests
Independent Samples
Tests the equality of the means from two independent groups (diagram below)
Relies on the t distribution to produce the probabilities used to test statistical significance
Dependent Samples
Tests the equality of the means between related groups or of two variables within the same group (diagram below)
Relies on the t distribution to produce the probabilities used to test statistical significance
Treatment group
Person #1
Control group
Person #2
Before treatment
Person #1
After treatment
Person #1
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47. Types One sample compare with population Unpaired compare with control
same subjects: pre-post
Z-test
large samples >30
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48. Compare Means (or medians)Example:
Compare blood presures of two or more groups, or compare BP of one group with a theoretical value.
1 Group:
One Sample t test
Wilcoxon rank sum test
2 Groups:
Unpaired t test
Paired t test
Mann-Whitney t test
Welch’s corrected t test
Wilcoxon matched pairs test
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49. Repeated measures ANOVA Kruskal-Wallis test Friedman test
3-26 Groups:
One-way ANOVA
Repeated measures ANOVA
Kruskal-Wallis test
Friedman test
(All with post tests) Raw data Average data Mean, SD, & NAverage data Mean, SEM, & N
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50. Statistical Analysis Is there a difference? control group mean
treatment
group
mean
Is there a difference?
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51. What does difference mean?
The mean difference
is the same for all
three cases
medium
variability
high
variability
low
variability
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52. What does difference mean?
medium
variability
high
variability
Which one shows
the greatest
difference?
low
variability
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53. t Test: Unknown Assumption
Population is normally distributed
If not normal, requires a large sample
T test statistic with n-1 degrees of freedom
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54. Example: One-Tail t Test
Does an average box of cereal contain more than 368 grams of cereal? A random sample of 36 boxes showed X = 372.5, and s = 15. Test at the a = level.
368 gm.
H0: m £ H1: m > 368
s is not given
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55. Example Solution: One-Tail
H0: m £ H1: m > 368
Test Statistic:
Decision:
Conclusion:
a = 0.01
n = 36, df = 35
Critical Value:
Reject
Do Not Reject at a = .01
.01
No evidence that true mean is more than 368
2.4377
t35
1.80
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56. The t Table
Since it takes into account the changing shape of the distribution as n increases, there is a separate curve for each sample size (or degrees of freedom).
However, there is not enough space in the table to put all of the different probabilities corresponding to each possible t score.
The t table lists commonly used critical regions (at popular alpha levels).
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57. Z-distribution versus t-distribution
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58. The z-Test for Comparing Population Means
Critical values for standard normal distribution
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59. Summary
We can use the z distribution for testing hypotheses involving one or two independent samples
To use z, the samples are independent and normally distributed
The sample size must be greater than 30
Population parameters must be known
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60. Thank you…!!!
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