1. Basics of statistics
Conducted by Dept. of Biostatistics, NIMHANS
From 28 to 30 Sept, 2015
"It is easy to lie with statistics,
But it is hard to tell the truth without statistics."
–Andrejs Dunkels
2. Topics covered
• Introduction
• Types of statistics
• Definitions
• Variable & Types
• Variable scales
• Description of data
• Distribution of sample &
population
• Measures of center, dispersion &
shape
• Properties of Normal distribution
• Testing of hypothesis
• Types of error
• Estimation of sample size
• Various tests to be used
• Central limit theorem
• Parametric tests- t-test, ANOVA,
Post Hoc, Correlation &
Regression
• Non Parametric tests
• Tests for categorical data
• Summary of tests to be used
• Qualitative vs Quantitative
research
• Qualitative research
• Software packages
3. Statistics
• Consists of a body of methods for collecting and analyzing data.
• It provides methods for-
– Design- planning and carrying out research studies
– Description- summarizing and exploring data
– Inference- making predictions & generalizing about phenomena
represented by data.
4. Types of statistics
• 2 major types of statistics
• Descriptive statistics- It consists of methods for organizing and
summarizing information.
– Includes- graphs, charts, tables & calculation of averages, percentiles
• Inferential statistics- It consists of methods for drawing and measuring the
reliability of conclusions about population based on information obtained.
– Includes- point estimation, interval estimation, hypothesis testing.
• Both are interrelated. Necessary to use methods of descriptive statistics to
organize and summarize the information obtained before methods of
inferential statistics can be used.
5. Population & Sample
• Basic concepts in statistics.
• Population- It is the collection of all individuals or items under
consideration in a statistical study
• Sample- It is the part of the population from which information is
collected.
• Population always represents the target of an investigation. We learn about
population by sampling from the collection.
6. • Parameters- used to summarize the features of the population under
investigation.
• Statistic- it describes a characteristics of the sample, which can then be
used to make inference about unknown parameters.
7. Variable & types
• Variable- a characteristic that varies from one person or thing to another.
• Types- Qualitative/ Quantitative, Discrete/ Continuous, Dependent/
Independent
• Qualitative data- the variable which yield non numerical data.
– Eg- sex, marital status, eye colour
• Quantitative data- the variables that yield numerical data
– Eg- height, weight, number of siblings.
8. • Discrete variable- the variable has only a countable number of distinct
possible values.
– Eg- number of car accidents, number of children
• Continuous variable- the variable has divisible unit.
– Eg- weight, length, temperature.
• Independent variable- variable is not dependent on other variable.
– Eg- age, sex.
• Dependent variable- depends on the independent variable.
– Eg- weight of a newborn, stress
9. Variable scales
• Variables can also be described according to the scale on which they are
defined.
• Nominal scale- the categories are merely names. They do not have a
natural order.
– Eg- male/female, yes/no
• Ordinal scale- the categories can be put in order. But the difference
between the two may not be same as other two.
– Eg- mild/ Moderate/ Severe.
10. • Interval scale- the differences between variables are comparable. The
variable does not has absolute zero.
– Eg- temperature, time
• Ratio scale- the variable has absolute zero as well as difference between
variables are comparable..
– Eg- stress using PSS, insomnia using ISI
• Nominal & Ordinal scales are used to describe Qualitative data.
• Interval & Ratio scales are used to describe Quantitative data.
11. Describing data
• Qualitative data-
– Frequency- number of observations falling into particular class/
category of the qualitative variable.
– Frequency distribution- table listing all classes & their frequencies.
– Graphical representation- Pie chart, Bar graph.
– Nominal data best displayed by pie chart
– Ordinal data best displayed by bar graph
12. • Quantitative data-
– Can be presented by a frequency distribution.
– If the discrete variable has a lot of different values, or if the data is a
continuous variable then data can be grouped into classes/ categories.
– Class interval- covers the range between maximum & minimum
values.
– Class limits- end points of class interval.
– Class frequency- number of observations in the data that belong to
each class interval.
– Usually presented as a Histogram or a Bar graph.
13. Population & Sample distribution
• Population distribution- frequency distribution of the population.
• Sample distribution- frequency distribution of the sample.
• Sample distribution is a blurry photo of the population distribution.
• As the sample size ↑, the sample distribution becomes closer representative
of the population distribution.
• Sample of population distribution can be summarized by describing its
shape (based on the graph).
• It can be Symmetric or Nonsymmetric/ Skewed to left/ right based on its
tail.
14. Properties of
Numerical data &
Measures
Central tendency
Mean
Median
Mode
Dispersion
Range
Interquartile
Range
Standard
Deviation
Shape
Skewness
Kurtosis
15. Measures of center
• Central tendency- In any distribution, majority of the observations pile up,
or cluster around in a particular region.
– Includes- Mean, Median & Mode.
• Mean- sum of observed values in a data divided by the number of
observations
• Median- observation in the data set that divides the data set into half.
• Mode- value of the data set which occurs with greatest frequency
• Mean & Median can be applied only to Quantitative data
• Mode can be used either to Qualitative or Quantitative data.
16. What to choose?
• Qualitative variable- Mode.
• Quantitative with symmetric distribution- Mean.
• Quantitative with skewed distribution- Median.
• Outlier- observation that falls far from the rest of the data. Mean gets
highly influenced by the outlier.
• We use sample mean, median & mode to estimate the population mean,
median & mode.
17. Measures of dispersion
• Dispersion- It is the spread/ variability of values about the measures of
central tendency. They quantify the variability of the distribution.
• Measures include-
– Range
– Sample interquartile range
– Standard deviation
• Mostly used for quantitative data
• Range- difference between the largest observed value in the data set and
the smallest one.
– So, while considering range great deal of information is ignored.
18. • Interquartile range- difference between the first & third quartiles of the
variable.
– Percentile- divides the observed values into hundredths/ 100 equal
parts.
– Deciles- divides the observed values into tenths/ 10 equal parts
– Quartiles- divides the observed values into 4 equal parts. Q1 divides
the bottom 25% of observed values from top 75%...
• Standard deviation- it is a kind of average of the absolute deviation of
observed values from the mean of the variable.
– It is defined using the sample mean & values get strongly affected by
few extreme observations.
19. Shape
• Skewness- Lack of symmetry in distribution. It can be interpreted from
frequency polygon.
• Properties-
– Mean, median & mode fall at different points.
– Quartiles are not equidistant from median.
– Curve is not symmetrical but stretched more to one side.
• Distribution may be positively or negatively skewed. Limits for
coefficient of skewness is ± 3.
• Kurtosis- convexity of a curve.
– Gives an idea about the flatness/ peakedness of the curve.
20. Normal distribution
• Bell shaped symmetric distribution.
• Why is it important?
– Many things are normally distributed, or very close to it.
– It is easy to work with mathematically
– Most inferential statistical methods make use of properties of the
normal distribution.
• Mean = Median = Mode
• 68.2% of the values lie within 1SD.
• 95.4% of the values lie within 2SD.
• 99.7% of the values lie within 3SD.
21. Tests to check normal distribution
1. Checking measures of Central tendency, Skewness & Kurtosis.
2. Graphical evaluation- normal plot, frequency polygon.
3. Statistical tests-
– Kolmogorov-Smirnov test
– Shapiro-Wilk test
– Lilliefor’s test
– Pearson’s chi-squared test
• Shapiro-Wilk has the best power for a given significance.
• If not normally distributed?- correction by transformation of the data- log
transformation, square root transformation.
22. Hypothesis testing
• Aim of doing a study is to check whether the data agree with certain
predictions. These predictions are called hypothesis.
• Hypothesis arise from the theory that drives the research.
• Significance test- it is a way of statistically testing a hypothesis by
comparing the data values.
– It consists of two hypothesis- Null (H0) & Alternative hypothesis (H1).
– Null hypothesis is usually a statement that the parameter has value
corresponding to, in some sense, no effect.
– Alternative hypothesis is a hypothesis contradicts null hypothesis.
– Hypothesis are formulated before collecting the data.
23. • Significance test analyzes the strength of sample evidence against the null
hypothesis.
• The test is conducted to investigate whether the data contradicts the null
hypothesis, suggesting alternative hypothesis is true.
• Test statistics- statistic calculated from the sample data to test the null
hypothesis.
• p-value- is the probability, if H0 were true, that the test statistic would fall
in this collection of values. The smaller the p-value, the more strongly
the data contradicts H0.
• When p-value ≤ 0.05, data sufficiently contradicts H0.
24. Types of error
• Type I/ α error- Rejecting true null hypothesis.
– We may conclude that difference is significant, when in fact there is no
real difference.
– It is popularly known as p-value. Maximum p-value allowed is called
as level of significance. Being serious p-value is kept low, mostly less
than 5% or p<0.05.
• Type II/ β error- Accepting false null hypothesis.
– We may conclude that difference is not significant, when in fact there is
real difference.
– It is also called as Power of the test & indicates sensitivity of the test.
• Not possible to reduce both type I & II, So α error is fixed at a tolerable
limit & β error is minimized by ↑ sample size.
25. Estimation of Sample size
• Small sample- fails to detect clinically important effects (lack of Power)
• Large sample- identify differences which has no clinical relevance.
• Calculation is based on (not included formulas)-
– Estimation of mean
– Estimation of proportions
– Comparison in two means
– Comparison in two proportions
• Checklist- level of significance, power, study design, statistical procedure.
• Minimum sample size required for statistical analysis- 50.
26. Basic theorem in statistics
• Central limit theorem-
– States that the distribution of the sum/ average of a large number of
independent, identically distributed variables will be approximately
normal.
• Why is this important?
– Basis of many statistical procedures.
27. Parametric tests
• These are statistical tests that makes assumptions about the parameters
(defining properties).
• Assumptions made are-
– Data follows normal distribution.
– Sample size is large enough for Central limit theorem to lead to
normality of averages.
– Data is not normal, but can be transformed.
• Some situations where data does not follow normal distribution-
– Outcome is an ordinal variable.
– Presence of definite outliers
– Outcome has clear limits of demarcation.
28. Tests to be used
Scale type Permissible statistics
Nominal Mode
Chi-Square test
Ordinal Mode/ Median
Interval
Mean, Standard
Deviation
t-test, ANOVA, Post hoc,
Correlation, Regression,
Ratio
One sample
t-Test
Independent
t- test
Dependent
t- test
Compares the sample
mean with the
population mean
Compares the means of
two independent
samples
Compares the means of
paired samples
(before-after, pre-post)
29. ANOVA
• t- Test- difference between 2 means.
– If there are more than 2 means, then doing t test increases the α & β
error. Which creates a serious flaw.
• So when there are >2 means to be compared we use ANOVA.
• Types-
– One way- study effects of one factors.
– Two way- study effects of multiple factors.
• Assumptions of ANOVA- Normality, Linearity.
• ANCOVA- It is a blend of ANOVA & Regression. In other words,
measures how much 2 variables change together & how strong is the
relationship.
30. Post Hoc
• Latin phrase, means- “after this” or “after the event”
• Why do Post hoc tests?
– ANOVA tells whether there is an overall difference between groups,
but it does not tell which specific group differed.
– Post hoc tests tell where the difference occurred between groups.
• Different Post hoc tests-
– Bonferroni
– Fisher’s least significant difference (LSD)
– Tukey’s honestly significant difference (HSD)
– Scheffe post hoc tests
31. Correlation & Regression
• Correlation- denotes association between 2 quantitative variables.
– Assume that the association is linear (i.e.., one variable ↑/ ↓ a fixed
amount for a unit ↑/ ↓ in the other).
– Degree of association is measured by a correlation coefficient, r.
– r is measured on a scale from -1 through 0 to +1.
– When both variables ↑, then r is + & when 1 variable ↑ and other
decreases, then r is -.
• Graphically- Scatter diagrams, usually independent variable is plotted
against x-axis & dependent against y-axis.
• Limitation- it does not say anything about Cause & Effect relationship.
– Beware of spurious/ non sense correlation.
32. • Correlation-
– Strength/ degree of association.
• Regression-
– Nature of association (eg- if x & y related, it means if x changes by
certain amount then y changes on an average by certain amount).
– Expresses the linear relationship between variables.
– Regression coefficient- β
– Types- Linear, Non linear, Stepwise
• Regression coefficient gives a better summary of the relationship between
the two variables than Correlation coefficient.
33. Non Parametric tests
• Also called as “Distribution free tests”, because they are based on fewer
assumptions.
• Advantages-
– When data does not follow normal distribution.
– When the average is better represented by median.
– Sample size is small.
– Presence of outliers.
– Relatively simple to conduct
34. Tests
Characters Parametric test Non Parametric test
Testing mean, a
hypothesized value
One sample t test Sign test
Comparison of means of
2 groups
Independent t test Mann Whitney U test
Means of related samples Paired t test
Wilcoxon Signed rank
test
Comparison of means of
> 2 groups
ANOVA Kruskal Wallis test
Comparison of means of
> 2 related groups
Repeated measures of
ANOVA
Friedman’s test
Assessing the
relationship between 2
quantitative variables
Pearson’s correlation Spearman’s correlation
35. Chi-Square test
• Used for analysis of categorical data.
• Other tests- Fisher exact probability test, McNemar’s test.
• Requirements of Chi-Square-
– Sample should be independent
– Sample size should be reasonably large (n >40)
– Expected cell frequency should not be < 5.
• Yate’s correction- if expected cell frequency is < 5
• Fisher exact probability test- used when sample size is small (n < 20)
• McNemar’s test- used when there are two related samples or there are
repeated measurements
36. RR & OR
• Relative Risk (RR)-
– It is the ratio of incidence rate among exposed to the incidence rate
among not exposed.
– used in RCTs & Cohort studies
– Values- <1 - risk of disease is less among exposed
– >1 – risk of disease is more among exposed
– =1 – equal risk among exposed & non exposed
• Odds Ratio (OR)-
– Ratio of odds of exposure among the cases to odds of exposure among
controls. Used for rare diseases/ events
– Used in case control & retrospective studies (no meaning in calculating
the risk of getting the disease)
– Values- >1- more among cases, <1- more among controls
37. Qualitative v/s Quantitative
Qualitative research
• Seeks to confirm hypothesis
• Highly structured methods used
• Uses closed ended, numerical
methods of collecting data
• Study design is fixed & subject to
statistical assumptions
Quantitative research
• Seeks to explore phenomena
• Semi-structured methods used
• Uses open ended, textual methods
• Study design is flexible, iterative
& subject to textual analysis
38. Qualitative research
• Provides complex descriptions & information about issues such as
contradictory behavior, belief, opinions, emotions & relationships.
• Methods used are-
– Phenomenology
– Ethnography
– Grounded theory
• Designs used-
– Case studies
– Comparative designs
– Snapshots
– Retrospective & Longitudinal studies
39. Statistical software packages
Quantitative research
• SPSS by IBM
• R by R Foundation
• GenStat by VSN International
• Mathematica by Wolfram
research
• Minitab, MATLAB, Nmath Stats
etc..,
Qualitative research
• ATLASti
• NVIVO
• MAXQDA
• NUDist
• ANTHTOPAC
40. "An approximate answer to the right problem is worth a good deal,
more than an exact answer to an approximate problem." -- John Tukey
