tests of significance in periodontics aspect, tests of significance with common examples, tests in brief, null hypothesis, parametric vs non parametric tests, seminar by sai lakshmi

1. TESTS
OF SIGNIFICANCE
PRESENTED BY:
DR. SAI LAKSHMI C
1ST YEAR PG

2. 2
• Historical aspect
• Basis of statistical inference
• Statistics
• Biostatistics
• Data
• Sample & Sampling
• Measures of Central tendency
• Measures of Dispersion
• Standard error
• Hypothesis testing
• Null hypothesis
• Alternate hypothesis
• Type I error and Type II error
• P- value
• Power of test
• Confidence level
• One-tailed test, Two-tailed test.
• Effect of sample size
• Tests of significance
• Parametric Vs Non Parametric test
• Parametric test
• Non-parametric test
• References

3. • Father of Statistics: Gottfried Achenwall
• Father of Biostatistics: Francis Galton
• Statistical significance: Ronald Fisher
• Student t-test: William Sealy Gosset
Gottfried Achenwall
William Sealy Gosset
Ronald Fisher

4. • Statistical inference is the branch of statistics which is
concerned with using probability concept to deal with
uncertainty in decision making.
• It refers to the process of selecting and using a sample to
draw inference about a population from which sample is
drawn.

5. Statistics is a field of study concerned with-
1- Collection, organization, summarization and analysis of
data
2- Drawing of inferences about a body of data when only a
part of the data is observed

6. • Statistical processes and methods applied to the collection, analysis,
and interpretation of biological data and especially data relating to
human biology, health, and medicine.
• Development and application of statistical techniques to scientific
research relating to life (humans, plants, animals, etc.) Here the
focus is on human health and life. Thus the areas of application
related to:-
Pharmacology, medicine, epidemiology, public health, physiology,
anatomy, and genetics

7. -The raw material of Statistics is data.
-Data is defined as figures
-Figures result from the process of counting or from taking a
measurement
For example:
• When a hospital administrator counts the number of
patients (counting)
• When a nurse weighs a patient (measurement)

8. 1- Routinely maintained records
For example:
Hospital medical records contain immense amounts of
information of patients
2- Questionnaires – in this method, a list of some questions
pertaining to survey, known as a ‘questionnaire’ is prepared and
the various informants are requested to supply the information
either personally or through posts.
3- Surveys

9. 4- Interviews – in this method, there is face-to-face interaction with
the person from whom the information is obtained.
Advantage:
-all the information can be collected accurately
Disadvantage:
-this method is time consuming and requires more personnels.

10. Quantitative Variables
It can be measured.
For example:
-Height
-Weight
Qualitative Variables
Many characteristics are not
capable of being measured.
Some are ordinal and some are
nominal.
For example:
-Gender
-Malocclusion

11. Continuous variable
It can assume any value within
a specified relevant interval of
values
For example:
-Height
-weight
Discrete variable
It is characterized by gaps or
interruptions in the values
that it can assume
For example:
- The number of decayed,
missing or filled teeth in an
elementary school child

12. Nominal
It consists of “naming” or classifies
into various mutually exclusive
categories
For example:
- Male/ female
- Sick/ well
- Married/ single/ divorced
Ordinal
Whenever qualitative observation
can be ranked or “ordered”
according to some criterion.
For example:
- Blood pressure (high / low)
- Grades (Excellent/ V. good/ good/
fail)

13. • A sample is a set of data chosen from a population and is a subset
of the population.
• A complete collection of all elements (scores, people, measurements,
etc.) to be studied. The collection is complete in the sense that it
includes all the subjects to be studied.

14. • Sampling can be defined as the investigation of part of a
population, in order to provide information, which can then
be generalized to cover the whole population.
POPULATION
SAMPLE
• Measuring a small portion of
something and then making a
general statement about the
whole thing.

15. The word average implies a value in the distribution, around which
the other values are distributed
It gives a mental picture of the central value
Commonly used averages:
1. Arithmetic Mean
2. Median
3. Mode

16. • Also known as arithmetic average or arithmetic mean
• The MEAN is the sum of a collection of numbers divided by the
count of numbers in the collection. The collection is often a set
of results of an experiment or an observational study, or
frequently a set of results from a survey
• Mean locate the center of distribution.
• Eg: fasting blood sugar level of 6 persons
1 2 3 4 5 6
62 63 66 68 67 64
Total 390
Mean 390/6 = 65

17. • Average of a different kind, which does not depend upon the total
and number of items.
• To obtain the median, the data is first arranged in an ascending value
or descending order of magnitude, and then the value of the middle
observation is located, which is called the “median”.
• Eg : No. of students in 5 classrooms
50 52 56 58 59;
(50, 52, 56, 58, 59); Median = 56
51 59 55 56 52 53;
(59, 56, 55, 53, 52, 51); median = (55 + 53) ÷ 2 = 54

18. • The most commonly occurring value in a distribution of data
• It is the most frequent item or the most fashionable value in a
series of observations
• Advantage: Easy to understand,
Not affected by the extreme items
• Eg: diastolic pressure of 5 individuals-
85 80 85 81 85; Mode = 85

19. • Study the spread of values about the central value
• Common measures of dispersion used are:
Range
Standard deviation
Co-efficient of variation
PURPOSE:
• To study the variability of data
• For accounting the variability in data

20. • Range:
Difference between the highest and the lowest figures in a
given sample.
• Standard Deviation:
It is the measure of spread of scores within a set of data.
• Mean deviation:
It is the average of the deviations from a central point.

21. • The standard error is a statistical term that measures the
accuracy with which a sample distribution represents a
population by using standard deviation.

22. Hypothesis testing refers to-
1. Making an assumption, called hypothesis, about a population
parameter
2. Collecting sample data
3. Calculating a sample statistic
4. Using the sample statistic to evaluate the hypothesis (how
likely is it that our hypothesized parameter is correct. To test
the validity of our assumption we determine the difference
between the hypothesized parameter value and the sample
value)
5. Hypothesis is a tentative justification, the validity of which
remains to be tested.
HYPOTHESIS & ITS TESTING

23. HYPOTHESIS
TESTING
NULL HYPOTHESIS,
H0
ALTERNATIVE
HYPOTHESIS, H1
• States the hypothesized value
of the parameter before
sampling.
• The assumption we wish to
test (or the assumption we
are trying to reject)
• E.g. There is no difference
between povidone iodine and
saline.
• All possible hypothesis other
than the null hypothesis
• E.g. There is a difference
between povidone iodine and
saline.

24. 1. Hypothesis should be clear and precise.
2. Hypothesis should be capable of being tested.
3. It should state relationship between variables.
4. It must be specific.
5. It should be stated as simple as possible.
6. It should be amenable to testing within a reasonable time.
7. It should be consistent with known facts.
CHARACTERISTICS OF HYPOTHESIS

25. A statement in which no difference or effect is expected. If
the null hypothesis is not rejected, no changes will be made.
The Null Hypothesis is symbolized as Ho and Alternative
Hypothesis is symbolized as H1 or HA.
In Hypothesis testing we proceed on the basis of Null
Hypothesis. We always keep Alternative Hypothesis in mind. The
Null Hypothesis and the Alternative Hypothesis are chosen before
the sample is drawn.
NULL HYPOTHESIS

26. • A Null Hypothesis (H0) or Hypothesis of no difference between
statistic of a sample and parameter of population or between
statistic of two samples nullifies the claim that the experimental
result is different from or better than the one observed already.
In other words, Null Hypothesis states that the observed
difference is entirely due to sampling error, that is - it has
occurred purely by chance.
• H0 asserts that there is no real difference in the sample and the
population under consideration, and the difference is accidental.
• Example: “there is no difference in the DMF scores of the rural
and urban children”.
NULL HYPOTHESIS

27. • Alternative hypothesis of significant difference states
that the sample result is different that is greater or
smaller than the hypothetical value of population
• A test of significance such as z-test, t-test, chi-squared
test, is performed to accept the null hypothesis or to
reject it and accept the alternative hypothesis.
• If our sample does not support this null hypothesis, we
should conclude that something else is true.
• What we conclude rejecting the null hypothesis…
• Example, “there is a difference in the DMF scores of the
rural and urban children.
ALTERNATIVE HYPOTHESIS

28. When a null hypothesis is tested, there may be 4 possible outcomes-
i. The Null Hypothesis is true but our test rejects it.
ii. The Null Hypothesis is false but our test accepts it.
iii. The Null Hypothesis is true and our test accepts it.
iv. The Null Hypothesis is false and our test rejects it.
TYPE 1 & TYPE 2 ERRORS

29. Type 1 Error – rejecting Null Hypothesis when Null Hypothesis
is true. It is called ‘α error’.
Type-I error occurs when the sample results, lead to the
rejection of the null hypothesis when it is in fact true. Type-
I errors are equivalent to false positives.
Type 2 Error – accepting Null Hypothesis when Null Hypothesis
is false. It is called ‘β-error’.
Type-II error occurs when based on the sample results, the null
hypothesis is not rejected when it is in fact false. Type-
II errors are equivalent to false negatives.

31. • P-value is the probability of obtaining results at least as
extreme as the observed results of a statistical hypothesis
test, assuming that the null hypothesis is correct.
• A smaller p-value means that there is stronger evidence in
favor of the alternative hypothesis.
P - VALUE

32. • The probability of committing Type 1 Error is called the p-
value.
• Thus p-value is the chance that the presence of difference is
concluded when actually there is none.
• When the p-value is between 0.05 and 0.01 the result is
usually called significant.
P - VALUE

33. • Power of a hypothesis test is the probability that the test rejects
the null hypothesis when a specific alternative hypothesis is true
• It indicates the probability of avoiding a type II error
• The statistical power of a test is the probability that a study or a
trial will be able to detect a specified difference
• This is calculated as 1- probability of type II error, i.e. probability
of correctly concluding that a difference exists when it is indeed
present. Thus, power = 1-β
POWER OF TEST

34. Confidence Interval: The interval within which a parameter value is
expected to lie with a certain confidence level as could be revealed by
repeated samples is called confidence interval.
Confidence Level: The degree of assurance for an interval to contain
the value of a parameter (1-α).
CONFIDENCE LEVEL

35. • A one-tailed test is a statistical test in which the critical
area of a distribution is one-sided so that it is either
greater than or less than a certain value, but not both.
• If the sample being tested falls into the one-sided critical
area, the alternative hypothesis will be accepted instead
of the null hypothesis.
• A one-tailed test is also known as a directional hypothesis
or directional test.
ONE-TAILED HYPOTHESIS TESTING

36. • Hypothesis testing is run to determine whether a claim is
true or not, given a population parameter.
• When the testing is set up to show that the sample mean
would be higher or lower than the population mean, it is
referred to as a one-tailed test.
• A one-tailed test is a statistical hypothesis test set up to
show that the sample mean would be higher or lower than
the population mean, but not both.
ONE – TAILED TEST

37. • A test that is conducted to show whether the mean of the sample is
significantly greater than and significantly less than the mean of a
population is considered a two-tailed test.
• It is used in null-hypothesis testing and testing for statistical
significance.
• If the sample being tested falls into either of the critical areas, the
alternative hypothesis is accepted instead of the null hypothesis.
• By convention, two-tailed tests are used to determine significance
at the 5% level, meaning each side of the distribution is cut at 2.5%.
TWO-TAILED HYPOTHESIS TESTING

39. • Your target sample size is how many people you need to reach to derive
accurate insights from your study.
• A study that has a sample size which is too small may produce
inconclusive results and could also be considered unethical, because
exposing human subjects or lab animals to the possible risks associated
with research is only justifiable if there is a realistic chance that the
study will yield useful information.
• Similarly, a study that has a sample size which is too large will waste
scarce resources and could expose more participants than necessary to
any related risk. Thus an appropriate determination of the sample size
used in a study is a crucial step in the design of a study.
EFFECT OF SAMPLE SIZE ON TEST

40. TESTS OF SIGNIFICANCE
• Whenever two sets of observations are compared, it becomes
essential to find out whether the difference observed between the
two groups is because of sampling variation or any other factor.
• In statistics, it is important to know if the result of an experiment
is significant enough or not. In order to measure the significance,
there are some predefined tests which could be applied. These
tests are called the tests of significance.

41. • Test of significance is a formal procedure for comparing
observed data with a claim (also called a hypothesis) whose
truth we want to assess.
• Test of significance is used to test a claim about an unknown
population parameter.
• A significance test uses data to evaluate a hypothesis by
comparing sample point estimates of parameters to values
predicted by the hypothesis.
• We answer a question such as, “If the hypothesis were true,
would it be unlikely to get data such as we obtained?”

42. • Test statistic is based on
the distribution
• Uses a mean value for
central tendency
• Requires previous
knowledge about the
population
• Parametric test is
powerful, if it exist
• Test statistic is arbitrary
• Uses a median value for
central tendency
• Doesn’t require previous
knowledge about the
population
• Non parametric tests do exist
for nominal and ordinal scale
data
• It is not so powerful like
parametric test
PARAMETRIC VS NON-PARAMETRIC TESTS

43. Parametric Tests
o Student’s t-test (one
sample, two sample,
and paired)
o Z test
o ANOVA F-test
o Pearson’s correlation(r)
Non-Parametric Tests
o Sign test
o Wilcoxon Signed-Rank test
o Wilcoxon Rank Sum test
o Chi-square test
o Spearman’s Rank
Correlation(p)
o ANOVA
o Kruskal-Wallis test

44. Purpose of
application
Parametric test Non-Parametric test
Comparison of two
independent groups.
‘t’-test for independent
samples
Wilcoxon rank sum test
Test the difference
between paired
observation
‘t’-test for paired
observation
Wilcoxon signed-rank
test
Comparison of several
groups
ANOVA Kruskal-Wallis test
Quantify linear
relationship between
two variables
Pearson’s Correlation Spearman’s Rank
Correlation
Test the association
between two qualitative
variables
_ Chi-square test
SUMMARY OF COMMONLY USED PARAMETRIC &
NON-PARAMETRIC TESTS

45. • Statistical tests are
intended to decide whether
a hypothesis about
distribution of one or more
populations or samples
should be rejected or
accepted.
Statistical
tests
Parametric
tests
Non-parametric
tests

46. • Parametric test is a statistical test that makes assumptions about
the parameters of the population distribution(s) from which one’s
data is drawn.
APPLICATION:
• Used for Quantitative data.
• Used for continuous variables.
• Used when data are measured on approximate interval or ratio
scales of measurement.
• Data should follow normal distribution.

47. PARAMETRIC TESTS
1. t-test
t-test
t-test for one
sample
Unpaired two
sample t-test
t-test for two
samples
Paired two
sample t-test

48. 2. ANOVA
4. Z test
ANOVA
One way ANOVA
Two way ANOVA

50. • A t-test is a type of inferential statistic used to determine if
there is a significant difference between the means of two
groups, which may be related in certain features.
• Used in hypothesis testing to determine whether a process or
treatment actually has an effect on the population of interest,
or whether two groups are different from one another.
T-TEST

51. • One sample– only one group is studied and an externally
determined claim is examined.
• Two sample– there are two groups to compare.
• Paired– used when two sets of measurements are
available, but they are paired.
TYPES OF STUDENT T-TEST

52. • A one sample t-test of means compares the mean of a sample to a
pre-specified value and tests for a deviation from that value. This
test is also known as:
-Single Sample t Test
• For example, we might know that the average birth weight for
white babies in the US is 7.5 lbs and wish to compare the average
birth weight of a sample of black babies to this value.
ONE SAMPLE STUDENT T-TEST

53. TWO SAMPLE STUDENT T-TEST
• A two-sample t-test is used to test the difference (d0) between
two population means.
• A two-sample t-
test is used when you want to compare two independent
groups to see if their means are different. Used when two
independent random samples come from the normal populations
having unknown or same variance.

54. • This test is used to determine whether there is a statistical evidence that
the mean difference between paired observations on a particular
outcome is significantly different from zero.
• The Paired Sample t Test is a parametric test.
• This test is also known as:
-Dependent t Test
-Repeated Measures t Test
The variable used in this test is known as:
-Dependent variable, or test variable (continuous), measured at two
different times or for two related conditions or units
PAIRED T-TEST

55. • For e.g. Thirty sets of identical twins were enrolled in a
study to measure the effect of home environment on certain
social attitudes. One twin in each set was randomly assigned
to a minority environment or a home environment.
• It is a Paired Experiment as the investigator has used sets of
twins. Typically when this is done the analysis will be based
on the differences between sets of scores rather than
differences between the averages of one group versus the
other.

56. Z-TEST
• Z-test is a statistical test where normal distribution is
applied and is basically used for dealing with problems
relating to large samples when the frequency is greater
than or equal to 30.
• It is used when population standard deviation is known.

57. ANOVA (Analysis of Variance)
• Analysis of Variance (ANOVA) is a collection of statistical
models used to analyse the differences between group means
or variances.
• Compares multiple groups at one time
• Developed by R.A.Fischer

58. ANOVA
Three Way ANOVA
Two Way ANOVA
One Way ANOVA
Effect of Age, SES and
Diet on BMI
Effect of Age and SES on
BMI
Effect of SES on BMI
ANOVA with repeated measures- comparing >3 group means where the
participants are same in each group.
E.g. group of subjects is measured more than twice generally over
time, such as patients weighed at baseline and every month after a
weight loss program.

59. One way ANOVA
• Compares two or more unmatched groups when data is
categorized in one factor
• E.g.
-Comparing a control group with three different doses of aspirin
-Comparing the productivity of three or more employees based on
working hours in a company

60. Two way ANOVA
• Used to determine the effect of two nominal predictor variables
on a continuous outcome variable.
• It analyses the effect of the independent variables on the
expected outcome along with their relationship to the
outcome itself.
Ex: Comparing the employee productivity based on the working
hours and working conditions.

61. • It is used by statisticians to determine whether there is a three-way
relationship among variables on an outcome.
• A three-way ANOVA tests which of three separate variables have an
effect on an outcome, and on how the variables’ effects interact with
one another.
• E.g. a pharmaceutical company may do a three-way ANOVA to determine
the effect of a drug on a medical condition. One factor would be the
drug, another may be the gender of the subject, and another may be the
ethnicity of the subject. This test allows the scientist to quantify the
effects of each and whether the factors interact.
Three way ANOVA

62. • Correlation is 'the relationship between two or more paired
factors or two or more sets.
• Correlation is a statistic that measures the degree to which two
variables move in relation to each other
• The degree of relationship is usually measured and represented
by a correlation coefficient.
• A correlation coefficient is numerical measure of the linear
relationship between two factors or sets of scores.
• Coefficient can be identified by either the letter r or the Greek
letter rho ().
PEARSON’S CORRELATION

63. 1. One sample test
• Chi-square test
• One sample sign test
2. Two samples test
• Median test
• Two samples sign test
3. K-samples test
• Median tets
• Kruskal Wallis test

64. • Chi-square test (X2):
– Used to compare between observed and expected data.
1. Test of goodness of fit
2. Test of independence
3. Test of homogeneity
• Kruskal-Wallis test:
– for testing whether samples originate from the same distribution.
– used for comparing more than two samples that are independent,
or not related
– Alternative to one way ANOVA.
• Wilcoxon signed-rank:
– used when comparing two related samples or repeated measurements on
a single sample to assess whether their population mean ranks differ.

65. • Median test:
– Use to test the null hypothesis that the medians of the populations
from which two samples are drawn are identical.
– The data in sample is assigned to two groups, one consisting of data
whose values are higher than the median value in the two groups
combined, and the other consisting of data whose values are at the
median or below
• Sign test:
– can be used to test the hypothesis that there is "no difference in
medians" between the continuous distributions of two random
variables X and Y,

66. • Chi-Square test is a statistical procedure used by researchers to examine
the differences between categorical variables in the same population
• Karl Pearson – 1900
• It is a non-parametric test not based on any assumption or distribution of
any variable.
• This statistical test follows a specific distribution known as chi-square
distribution
• The test which is used to measure the differences between what is observed
and what is expected according to an assumed hypothesis is called the chi-
square test.

67. • The Chi-square test is one of the most commonly used non-
parametric tests, in which the sampling distribution of the
test statistic is a chi-square distribution, when the null
hypothesis is true.
• The Greek Letter X
2 is used to denote this test.
• It can be applied when there are few or no assumptions
about the population parameter.
• Used to evaluate unpaired/unrelated samples and proportions

69. What is it used for?
• It is a non parametric statistical test that compares two paired groups,
and comes in two versions- the rank sum test or the signed rank test.
• The goal of the test is to determine if two or more sets of pairs are
different from one another in a statistically significant manner.
• It is used for population data that can be ranked but do not have
numerical values, such as customer satisfaction or music reviews.
• Non parametric tests do not have parameters and cannot be defined by
an equation as parametric distributions can.

70. The types of questions that the Wilcoxon Test can help us answer include
things like:
• Are test scores different from 5th grade to 5th grade for the same
students?
• Does a particular drug have an effect on health when tested on the same
individuals?
These models assume that the data comes from two matched, or
dependent, populations, following the same person or stock through time
or place.
The data is also assumed to be continuous as opposed to discrete.
Because it is a non-parametric test it does not require a particular
probability distribution of the dependent variable in the analysis.

71. • For paired data
• It is a non parametric test based on signs (positive and negative) of
the differences in the levels seen before and after a therapy .
The sign test is a statistical method to test for consistent differences
between pairs of observations.
Given pairs of observations for each subject, the sign test determines
if one member of the pair tends to be greater than (or less than) the
other member of the pair.
71
SIGN TEST

72. • For matched pairs.
• Used to compare two related samples, matched samples, or
repeated measurements on a single sample to assess whether their
population mean ranks differ (i.e. it is a paired difference test)
• It is a better test than the sign test– assigns rank to the differences
of n pairs after ignoring the + or – signs.
• The lowest difference gets rank 1 and the highest gets rank n.
WILCOXON SIGNED RANK TEST

73. • For unpaired two sample situation.
• If there are n1 subjects in the first sample and n2 in the
second sample, these (n1+n2) values are jointly ranked
from 1 to (n1+n2)
• {the sum of these ranks is obtained for those subjects only
who are in smaller group}.
WILCOXON RANK SUM TEST

74. • Spearman’s correlation is designed to measure the relationship
between variables measured on an ordinal scale of measurement.
• For example, it is used to evaluate whether the order in which
employees complete a test exercise is related to the number of
months they have been employed.
SPEARMAN RANK CORRELATION

75. Where,
di is the difference between the paired ranks
n is the number of pairs.
The Spearman rank correlation coefficient may lie between -1
to +1. Values close to +/-1 indicate a high correlation ; values
close to zero indicate lack of relationship.
6
2
n
i
N ( N 2
1)
d
  1    i1

76. • The Kruskal Wallis test is the non parametric alternative to the One
Way ANOVA.
• It is sometimes called the one-way ANOVA on ranks, as the ranks of the
data values are used in the test rather than the actual data points.
• The test determines whether the medians of two or more groups are
different.
• The Kruskal Wallis test will tell you if there is a significant
difference between groups. However, it won’t tell you which groups
are different
KRUSKAL-WALLIS TEST

77. • A medical researcher would like to investigate an anecdotal evidence that
certain anti-depressive drugs can have a positive side-effect of lowering
neurological pain in individuals with chronic, neurological back pain, when
administered in doses lower than those prescribed for depression. The researcher
identifies 3 well-known, anti-depressive drugs which might have this positive side
effect, and labels them Drug A, Drug B and Drug C. The researcher then recruits
a group of 60 individuals with a similar level of back pain and randomly assigns
them to one of three groups and prescribes the relevant drug for a 4 week
period. At the end of the 4 week period, the researcher asks the participants to
rate their back pain on a scale of 1 to 10 (10 = greatest level of pain). The
researcher wants to compare the levels of pain experienced by the different
groups at the end of the drug treatment period. The researcher runs a Kruskal-
Wallis H test to compare this ordinal, dependent measure (Pain Score) between
the three drug treatments (i.e., the independent variable, is the type of drug
with more than two groups).

78. • A statistical assessment of data collected is the most accurate
tool for coming to quick conclusions in any conducted study
design
• Statistical principles have been used in design and execution
of medical research projects for many years but workers in
dental health have, perhaps only more recently recognized the
benefits to be derived from their use
• So as a dental professionals its our duty to update our
knowledge in bio-statistical field to carry out better studies
and research

79. Essentials of Public Health Dentistry, 6th edition, Soben Peter
Introduction to biostatistics and research methods, 5th edition,
P.S.S. Sundar Rao
Applied statistics in Health Sciences, 2nd edition, NSN Rao
& NS Murthy, Jaypee
www.wikipedia.org