2. • Background
• Different versions of t-test
• Main usage of t-test
• t-test v/s z-test
• Assumptions of t-test
• Examples
3. • Introduced in 1908 by William Sealy Gosset.
• Gosset published his mathematical work under the pseudonym “Student”.
• Definition of t test: “ It’s a method of testing hypothesis about the mean of
small sample drawn from a normally distributed population when the
standard deviation for the sample is unknown.”
4. • Dependent variables are interval or ratio.
• The population from which samples are drawn is normally distributed.
• Samples are randomly selected.
• The groups have equal variance (Homogeneity of variance).
• The t-statistic is robust (it is reasonably reliable even if assumptions arenot
fully met.)
5. • The calculation of a confidence interval for a sample mean.
• To test whether a sample mean is different from a hypothesized value.
• To compare mean of two samples.
• To compare two sample means by group.
6. • Single sample t test – we have only 1 group; want to test against a
hypothetical mean.
• Independent samples t test – we have 2 means, 2 groups; no relation
between groups, Eg: When we want to compare the mean of T/T group
with Placebo group.
• Paired t test – It consists of samples of matched pairs of similar units or one
group of units tested twice. Eg: Difference of mean pre & post drug
intervention.
7. • It is used in measuring whether a sample value significantly differs from a
hypothesized value.
• For example, a research scholar might hypothesize that on an average it
takes 3 minutes for people to drink a standard cup of coffee.
• He conducts an experiment and measures how long it takes his subjects to
drink a standard cup of coffee.
• The one sample t-test measures whether the mean amount of time it took
the experimental group to complete the task varies significantly from the
hypothesized 3 minutes value.
8.
9. • 10 individuals had taken an exam and we want to test whether their
scores, all together, are significantly different from the score of 100.
•We need to calculate the degrees of freedom.
•Here, the degrees of freedom is simply the sample size minus one.
•Therefore, Degrees of freedom = n – 1 = 10 – 1 = 9
•Now, we will refer to a t table to determine the critical t value for 9 degrees
of freedom at the .05 level of significance.
•Looking at a t table, this value is 2.26 .
•Since our calculated t value of 4.61 is greater than the critical t value of
2.26, we can say that the scores of our sample of 10 individuals differ
significantly from the score of 100.
10.
11. •
• The independent sample t-test consists of tests that compare mean value(s)
of continuous-level (interval or ratio data), in a normally distributeddata.
• The independent sample t-test compares two means.
• The independent samples t-test is also called unpaired t-test/ two sample t
test.
• It is the t-test to be used when two separate independent and identically
distributed variables are measured.
• Eg: 1. Comparision of quality of life improved for patients who tookdrug
Valporate as opposed to patients who took drug Levetiracetam in
myoclonic seizures.
2.Comparasion of mean cholesterol levels in treatment groupwith
placebo group after administration of test drug.
12. • A random sample of each population isused.
• The random samples are each made up of independent observation.
• Each sample is independent of one another.
•The population distribution of each population must be nearly normal, or the
size of the sample is large.
13. To test the null hypothesis that the two population means, μ1 and μ2, are
equal:
• 1. Calculate the difference between the two sample means, x ̄1 − x 2̄.
• 2. Calculate the pooled standard deviation: sp
• 3. Calculate the standard error of the difference between the means:
• 4. Calculate the T-statistic, which is given by T = x 1̄ − x 2̄/S E (x ̄ 1 − x ̄ 2 )
• This statistic follows a t-distribution with n1 + n2 − 2 degrees of freedom.
• 5. Use tables of the t-distribution to compare your value for T to the t n1+n2−2
distribution. This will give the p-value for the unpairedt-test.
15. • Suppose we have to compare the mean value of two groups, one with 7
subjects and the other with 5 subjects .
• These were their scores:
For an independent or between subjects’
t test: df = n1+ n2 - 2
•Now, take the absolute value of this, which is 0.44.
•Now, for the .05 probability level with 10 degrees of freedom, we see from the
table that the critical t score is 2.228 for a two-tailedtest.
•Since the calculated t score is lower than the critical t score, the results arenot
significant at the .05 probability level.
18. • Suppose a sample of “n” subjects were given an antihypertensive drug we
want to check blood pressure before and after treatment . We want to find
out the effectiveness of the treatment by comparing mean pre & postt/t.
• To test the null hypothesis that the true mean difference is zero, the
procedure is as follows:
1.Calculate the difference (di = yi − xi) between the two observations on each
pair.
2.Calculate the mean difference, d.
3.Calculate the standard error of the mean differences.S.E=S.D/√n
4. Calculate the t-statistic, which is given by T = d/S.E, Under thenull
hypothesis, this statistic follows a t-distribution with n − 1 degrees offreedom.
5. Use tables of the t-distribution to compare your value for T to the tn−1
distribution. This will give the p-value for the pairedt-test.
19. • A z-test is a statistical test used to determine whether two population meansare
different when the variances are known and the sample size is large. The test
statistic is assumed to have a normal distribution with a known S.D.
• The z-test is a hypothesis test in which the z-statistic follows a normal
distribution.
• The z-test is best used for greater than 30 samples because, under the central
limit theorem, as the number of samples gets larger, the samples are considered
to be approximately normally distributed. When conducting a z-test, the null
and alternative hypothesis, alpha and z-score should be stated.
• Next, the test statistic should be calculated, and the results and conclusion
stated.
• z-test is used to test hypotheses about means for large samples (N>100) with a
known variance,We use t-test when the sample size is small (N < 100) and the
population variance is unknown.
• Ex: Comparing the prevelance of disease in men versus women.
20.
21. What is sample size?
•It is the total representative samples from
the given population used for the research
study.
22. Why only representative samples?
•Difficult to subject the entire
population for the research because
of
Economical reason
Ethical reason
Time constraints
23. •Should be optimal, neither high nor less.
•High
Study will become costly and time consuming
Unethical to include participants than the
required numbers.
•Less
Validity of the study is lost.
Low power of the study and the research
outcome is not trustable.
24. •The power is the probability to reject the null
hypothesis (Ho), given that the null hypothesis is
false.
•Minimum expected probability is 80%.
•It is applicable to those studies wherein there is null
and alternative hypothesis.
•The prevalence studies do not have any hypotheses
and thus power of the study is not applicable.
25. Power is determined by?
•The power of the study depends on
•Sample size
•Alpha level set in the study (it is
optimal to set at 5% [0.05] in medical
research)
•Effect size
26. Sample size and power
•Increase in sample size increases the power
of the study.
•80% and above of power is considered to be
enough in medical research.
•During sample size calculation, it is
mandatory to calculate the expected power
of the study to justify your sample size.
27. •Different formulae based on the study design
•Data obtained from previous similar studies are
used in the formula to determine the sample
size.
•If there is no similar studies, the researcher has
to perform a pilot study to obtain the values that
can be used for sample size calculation.
28.
29. For the level of confidence of 95%, which is conventional, Z value is 1.96.
*Daniel WW (1999). Biostatistics: A Foundation for Analysis in the Health Sciences.
7thedition. New York: John Wiley & Sons.
30. • This sample size formula is valid if the calculated sample
size is smaller than or equal to 5% of the population size
• Sample size calculation without considering total
population size should be avoided because:
It may overshoot the 5% of total population size.
Sometimes it may even overshoot the total
population size.
31. *Daniel WW (1999). Biostatistics: A Foundation for Analysis in the Health Sciences.
7thedition. New York: John Wiley & Sons.
32.
33. Formula for a binary outcome and equal sample sizes in both groups, assuming: alpha = 0.05 and
power =0.80 (beta = 0.20).
34. • When the significance level alpha is chosen at 0.05 one should enter
the value 1.96 for a in the formula.
• When beta is chosen at 0.20, the value 0.842 should be filled in for b
in the formula.
35. • X in the formula is the minimal difference between
the groups that the investigator considers biologically
plausible and clinically relevant.
36.
37. Formula for a continuous outcome and equal sample sizes in both groups, assuming: alpha
= 0.05 and power = 0.80 (beta = 0.20)
38. • There are many sample size estimation
formulae.
• Many software are available to determine the sample
size which has the advantage of calculating the
expected power as well.
39. Ethics
–moral principles of right and wrong
–not absolute; may vary by person, by time, by
place
–and may be in competition with each other
Research ethics
–incorporating ethical principles into research
practice
–may involve a balance between and within
principles and practices
–all stages, all those involved, from inception of
research through to completion and
publication of results and beyond
40. • 1947- Noremberg Code
• 1968- Helsinki Declaration
• 1979- Belmont Report
• 1993- CIOMS
• 2005- UNESCO
41. 1. Research should be designed, reviewed and
undertaken to ensure integrity and quality
2. Research staff and subjects must be informed
fully about the purpose, methods and
intended possible uses of the research, what
their participation in the research entails and
what risks if any, are involved.
Exceptionally, some variation may be acceptable
3. The confidentiality of information supplied by
research subjects and the anonymity of
respondents must be respected.
42. 4. Research participants must participate in a
voluntary way, free from any coercion.
Exceptionally, covert research and deception may be
acceptable.
5. Harm to participants must be avoided.
avoidance of harm extends to family, kin, community
groups should not be unreasonably excluded from
research
exceptionally, some limited short term and minimal harm
may be acceptable
6 The independence of the research must be
clear; any conflicts of interest or partiality
must be explicit.
43. • In social science research risks are diverse
Not only - potential physical or psychological harm;
discomfort or stress But also disruption or damage
to e.g.
a subject’s rights and dignity
a subject’s personal social standing
individual privacy
personal values and beliefs
a subject’s links to family and wider community
a subject’s occupational status or position
implications of revealing illegal, sexual or deviant
behaviour
…. as individuals, as whole communities, or
categories of people
44. • Voluntary Participation (Informed consent-
Components, deception – procedures)
• No harm to the subjects - Non-maleficence - Do no
harm (commission or omission) minimize harm
• PAC: PRIVACY, ANONYMITY AND CONFIDENTIALITY
• Beneficence - promotion of well being (maximize
benefit)
• Autonomy - make own decisions
• Integrity
45. • Subjects must agree to reveal information
about themselves.
• Subjects must be able to provide informed
consent.
• Behavior observed in public settings is assumed
to imply agreement to being observed.
• Subjects contacted after being observed in a
public setting must be informed they were
observed in a public setting.
46. • Subjects must be free from reasonably
anticipated physical or emotional harm.
• Subjects must be informed of the manifest
content of the information they will be asked
to reveal about themselves.
• It is permissible to deceive subjects, as long as
the deception cannot be anticipated to create
physical or emotional harm.
47. • Purpose of study.
• How respondent was selected.
• Results will be used for research and [other].
• Voluntary participation in the study or any
part of it.
• Respondent can keep any incentives if they
withdraw from the study.
• Confidentiality of responses.
• Contact information of the researcher.
48. Research which is deliberately opposed to the
interests of the research subjects
– may have negative impact on some subjects
Research which balances short-term risks to
subjects against longer terms gains to
beneficiaries
49. must not be undertaken lightly or routinely – only as
a last resort but may be justified
where it provides unique forms of evidence
where overt observation might alter the
phenomenon being studied
if important or significant issues are being
addressed, and matters of social significance are
being discovered which cannot be uncovered in
other ways
where there might be risks for participant or
researcher.
Would always require full review by R.E.C.
50. 1. Plagiarism
2. Fabrication and falsification
3. Non-publication of data
4. Faulty data-gathering procedures
5. Poor data storage and retention
6. Misleading authorship
7. Sneaky publication practices
51. • Sometimes called “cooking data”
• Data not included in results because they don’t
support the desired outcome
• Some data are “bad” data
• Bad data should be recognized while it is being
collected or analyzed
• Outlier – unrepresentative score; a score that
lies outside of the normal scores
• How should outliers be handled?
52. • Collecting data from participants who are not
complying with requirements of the study
• Using faulty equipment
• Treating participants inappropriately
• Recording data incorrectly
• Most important and most aggravating.
• Always drop non-compliers.
• Fix broken equipment.
• Treat subjects with respect and dignity.
• Record data accurately.
• Store data in a safe and private place for 3 years.
53. Misleading authorship—who should be an author?
– Technicians do not necessarily become joint authors.
– Authorship should involve only those who contribute
directly.
– Discuss authorship before the project!
• Publication of the thesis or dissertation
– Should be regarded as the student’s work
– Committee chair and members may be listed as secondary
authors
• Dual publication – a manuscript should only be
published in a single journal.
