Lecture notes HFS 3283 Adapted from Dr Naing USM

1. KNOWLEDGE FOR THE BENEFIT OF HUMANITYKNOWLEDGE FOR THE BENEFIT OF HUMANITY
BIOSTATISTICS (HFS3283)
INFERENTIAL STATISTICS
Dr.Dr. MohdMohd RazifRazif ShahrilShahril
School of Nutrition & DieteticsSchool of Nutrition & Dietetics
Faculty of Health SciencesFaculty of Health Sciences
UniversitiUniversiti SultanSultan ZainalZainal AbidinAbidin
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2. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Topic Learning Outcomes
At the end of this lecture, students should be able to;
• define inferential statistics.
• explain hypothesis tests, p value and type I and II error
• explain how to interpret confidence interval
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3. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
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4. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
What is INFERENTIAL STATISTICS?
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• It is the Statistical Technique/ method used to
infer the result of the sample (statistic) to the
population (parameter)
PopulationPopulation SampleSample

5. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Two types of inferential statistics
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• Hypothesis tests
– e.g.
• Comparing 2 means
• Comparing 2 proportions
• Association between one variable and another variable
• Estimation (Confidence interval)
– e.g.
• Estimating a mean (numerical)
• Estimating a proportion (categorical)
•
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6. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
HypothesisHypothesis
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• A statement derived from primary research
question(s)
• There may be more than one hypothesis to be
tested in one study, but the fewer the better.
• It usually comes out of a hunch, an educated
guess based on published results or preliminary
observations
• In writing a hypothesis;
– State hypothesis (hypotheses) clearly and specifically
– Include study and outcome factors
•

7. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Research question??
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• A statement derived from primary research
question(s)
• Most studies are concerned with answering the
following questions:
– What is the magnitude of a health problem or health
factor?
– What is the causal relation between one factor (or
factors) and the disease or outcome of interest?
– What is the efficacy of an intervention?
•

8. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Hypothesis testing
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• In Hypothesis testing, we answer to a specific
question related to a population parameter (e.g.
population mean nutrition knowledge score)
using a sample statistic (e.g. sample mean
nutrition knowledge score).

9. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Hypothesis testing
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• In Hypothesis testing, we answer to a specific
question related to a population parameter (e.g.
population mean BMI) using a sample statistic
(e.g. sample mean BMI).
• RQ – Is the BMI of the population different
from 24.5 kg/m2 or not?
• Answer – Yes or No
– Null hypothesis (Ho ; µ = 24.5)
– Alternate hypothesis (Ha ; µ ≠ 24.5)

10. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Null hypothesis
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• Is a mathematical statement of equality stated
before data collection or data analysis
• After statistical analysis, a p value is calculated.
• Based on this p value, decision is made to
either accept or reject the Ho.

11. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Null hypothesis (cont.)
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12. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Hypothesis testing (cont.)
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• RQ - Is the BMI of the population different
from 24.5 kg/m2 or not?
– Ho ; µ = 24.5 mean = 26.1, SD = 4.3
– Ha ; µ ≠ 24.5 n = 130
• This test is called “One sample t test”.
• At the end of the hypothesis testing, we will get a
P value.
– If the P value is < 0.05, we reject the Null Hypothesis
(Ho). And conclude as Ha.
– If the P value is ≥ 0.05, we cannot reject the Null
Hypothesis (Ho). And conclude as Ho.

13. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Hypothesis testing (cont.)
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• RQ - Is the BMI of the population different
from 24.5 kg/m2 or not?
– Ho ; µ = 24.5 mean = 26.1, SD = 4.3
– Ha ; µ ≠ 24.5 n = 130
• In above example, if we get P = 0.01, we reject the
null hypothesis (Ho), then ...
– We conclude as Alternative Hypothesis (Ha) … “the mean
BMI of the population is different from 24.5 kg/m2”.
– Alternatively, we may report as … “the mean BMI is
significantly different from 24.5 kg/m2”.
• Note:
• (1) The second conclusion is more commonly used in the literature.
• (2) These are “statistical conclusion”, not yet “research conclusion”.

14. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Hypothesis testing (cont.)
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• RQ - Is the BMI of the population different
from 24.5 kg/m2 or not?
– Ho ; µ = 24.5 mean = 26.1, SD = 4.3
– Ha ; µ ≠ 24.5 n = 130
• In above example, if we get P = 0.08, we CANNOT
reject the null hypothesis, then …
– We conclude as … “the mean BMI in the population is NOT
different from 24.5 kg/m2”.
– Alternatively we may report as ... “the mean BMI is NOT
significantly different from 24.5 kg/m2”.

15. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Hypothesis testing (cont.)
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• RQ - Is the BMI of the population different
from 24.5 kg/m2 or not?
– Ho ; µ = 24.5 mean = 26.1, SD = 4.3
– Ha ; µ ≠ 24.5 n = 130
• In this one sample t test, there are assumptions or
requirements that we need to fulfil/check.
– (1) The sample is selected by using random sampling.
– (2) Observations are independent.
– (3) The data is “normally distributed” (called “Normality
Assumption”).

16. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Steps in Hypothesis Testing
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Step (1):Step (1): Generate null and alternative hypothesis
Step (2):Step (2): Set the significance level (α)
Step (3):Step (3): Decide which statistical test to use and check
the assumptions of the test
Step (4):Step (4): Perform test statistic and associated p -value
Step (5):Step (5): Make interpretation (based on p value & CI)
Step (6):Step (6): Draw conclusion

17. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
P value?
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• The p value is the probability (likehood) that the
result or difference was due to chance.
• A p value of 0.05 indicates that a 5% probability
that the difference observed between the groups
was due to chance

18. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
P value? (cont.)
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• RQ - Is the BMI of the population different
from 24.5 kg/m2 or not?
– Ho ; µ = 24.5 mean = 26.1, SD = 4.3
– Ha ; µ ≠ 24.5 n = 130
• If the P value < 0.05, we reject the Null Hypothesis.
• P value is the probability of error if you reject the Null
Hypothesis and conclude as the Alternative Hypothesis.
• Example: P value = 0.01. It means that …
– There is 1% probability of error in our conclusion, if we conclude
as Alternative Hypothesis (“significantly different”).
– We normally, allow less than 5% error. That is why the cut-off
point for P value is 0.05.

19. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
P value? (cont.)
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• RQ - Is the BMI of the population different
from 24.5 kg/m2 or not?
– Ho ; µ = 24.5 mean = 26.1, SD = 4.3
– Ha ; µ ≠ 24.5 n = 130
• If the P value < 0.05, we reject the Null Hypothesis.
• P value is the probability of error if you reject the Null
Hypothesis and conclude as the Alternative Hypothesis.
• Example: P value = 0.2. It means that …
– There is 20% probability of error in our conclusion, if we
conclude as Alternative Hypothesis (“significantly different”).
– Therefore, we can’t conclude as it is “significantly different”.
We have to conclude as “the difference is not significant”.

20. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
P value? (cont.)
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• RQ - Is the BMI of the population different
from 24.5 kg/m2 or not?
– Ho ; µ = 24.5 mean = 26.1, SD = 4.3
– Ha ; µ ≠ 24.5 n = 130
• If the P value < 0.05, we reject the Null Hypothesis.
• It means that we have set the cut-off point at P less than
0.05 to reject the Ho.
• We say this as setting the “Alpha” at 0.05
• Because the type of error that we have been talking
about, is called “Type I error” or “Alpha error”.

21. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
One-sided or Two-sided test?
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• RQa - Is the BMI of the population different
from 24.5 kg/m2 or not?
– Ho ; µ = 24.5
– Ha ; µ ≠ 24.5
• RQb - Is the BMI of the population more than
24.5 kgm2 or not?
– Ho ; µ ≤ 24.5
– Ha ; µ > 24.5
• RQc - Is the BMI of the population less than
24.5 kgm2 or not?
– Ho ; µ ≥ 24.5
– Ha ; µ < 24.5
TwoTwo--sided hypothesis testsided hypothesis test
OneOne--sided hypothesis testsided hypothesis test
OneOne--sided hypothesis testsided hypothesis test

22. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
One-sided or Two-sided test? (cont.)
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23. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Type I (α) and Type II (ɞ) Error
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Usually we set;
• Type I (α) error
= 5%
• Type II (ɞ) error
= 20%

24. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Type I (α) and Type II (ɞ) Error (cont.)
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Power = 1 - ɞ
Usually, power = 1 – 0.2
= 0.8 or 80%

25. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Type I (α) and Type II (ɞ) Error (cont.)
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26. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Estimation (Confidence Interval)Estimation (Confidence Interval)
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• Used to express degree of confidence in an
estimate
• Denotes the range of possible values the
estimate may assume, with a certain degree of
assurance
• Gives reliability on an estimate.
• E.g.
– If mean systolic BP level is 118 mmHg and 95% CI is
(110, 125); it means that you can be 95% confident
that the true systolic BP level mean lies between
110mmHg and 125mmHg.

27. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Estimation a Mean
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• Formula = mean ± (t α/2 x SE)
• Calculation example;
– The systolic BP of 100 students in a class
Mean = 123.4 mmHg
SD = 14.0 mmHg
– The 95% CI = Mean ± t α/2 x (SD/√n)
= 123.4 ± (1.98 x 14/10)
– The 95% CI ranges from 120.6 mmHg to 126.2 mmHg
– We can be 95% sure that this range includes the true
population mean.
•

28. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Assumption or Requirement
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Calculation can be made only ...
1) When the sample is selected by using random
sampling
2) When the observations are independent
3) When the data is “normally distributed” (called
“Normality Assumption”)

29. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Estimation a Proportion
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• Formula = p ± (Z α/2 x SE)
• Calculation example;
– The abnormal systolic BP of 100 students in a class
Prevalence = 37%
– The 95% CI = p ± (Z α/2 x SE)
= 0.37 ± (1.96 x √((0.37*0.63)/100)
– The 95% CI ranges from 0.27 to 0.47 (or 27% to
47%)
– We can be 95% sure that this range includes the
true population proportion.
•

30. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Assumption or Requirement
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Calculation can be made only ...
1) When the sample is selected by using random
sampling
2) When the observations are independent
3) BOTH (n*p) and (n *(1-p)) must be more than 5
count.
Note:
• Recode into 0 and 1;
– 1 should be ‘disease’ or condition of interest.
– 0 should be ‘non-disease’ or the rest.

31. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
P value and Confidence Interval
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• When using statistics to compare 2 groups, 2
approaches can be used;
– Calculation of p value
– Calculation of confidence interval
• Both p values and confidence interval are
complimentary and should be used together.

32. S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N
Limitation of P value
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• p values are sensitive to sample size
– Large sample size, smaller p value
• p values are sensitive to the magnitude of
difference between the two group:
– If the difference is small but samples are large, results
can still be statistically significant
– If the difference is large but samples are small, result
would not be statistically significant

33. Thank YouThank You
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