Statistics for Anaesthesiologists covers basic to intermediate level statistics for researchers especially commonly used study designs or tests in Anaesthesiology research.

1. Statistics for Anaesthesiologists
Dr John George K. MD,PDCC
Associate Professor of Anaesthesiology
KMC, Manipal

2. Recommended Software
• RStudio (GUI) with R, R Commander, R Commander
Plugins like EZR (Free, Cross platform, powerful
programming paradigm)
• G*Power (Free, for power analysis)
• SPSS (Commercial, expensive)
• SOFA (Free, basic)
• Graphpad.com
• Spreadsheet software like MS Excel for initial data entry
(export as CSV file format)

3. Data Types
• Nominal or Categorical data
• Ordinal data
• Interval data
• Ratio data

4. Data Types
 Nominal: Categorical data and numbers that are simply used
as identifiers or names. Ex: social security (Aadhar) number
 Ordinal: an ordered series of relationships or rank order. Ex:
first, second, or third place in a contest, Likert scale
 Interval: A scale that represents quantity and has equal units
but for which zero represents simply an additional point of
measurement.. Ex: Fahrenheit scale
 Ratio: similar to the interval scale. However, this scale also has
an absolute zero (no numbers exist below zero). Ex: Height,
Weight

5. Parametric tests

6. Non-parametric
tests

7. Reporting data types
OK to compute
Nominal
Ordinal
Interval
Ratio
Frequency
Distribution
Yes
Yes
Yes
Yes
Median,
percentiles
No
Yes
Yes
Yes
Mean, SD, SE of No
mean
No
Yes
Yes
Ratio or
coefficient of
variation
No
No
Yes
No

8. Tests for normality of data
• Kolmogorov-Smirnov Test – inferior to others,
relies on goodness of fit of a sample with a normal
distribution curve, avoid its use!
• Shapiro-Wilk Test – better, mores specific, more
powerful especially with small sample sizes,
available in Rcommander, SPSS (under menu
Analyze>Descriptive Statistics>Explore)

9. Tests for normality of data
• D'Agostino-Pearson test
• Anderson-Darling test
• Q-Q (Quantile Probability) Plot – visual guide
• Histogram – inferior, look for Skew or Kurtosis
• Density Plot – better, look for Skew or Kurtosis

10. Choosing a statistical test
• Make sure you have adequate sample size (power)
to reject null hypothesis (Ho)
• Check is it one (only < or > μ, only one direction)
or two-tailed comparison (≠μ , test significance at
both sides) – in general use 2
• Look at your data types – ordinal, interval etc
• Do descriptive statistics testing

11. Choosing a statistical test
• Test normality of data – tests and visual
comparison (especially when n<30)
• Decide to use Parametric Vs Non-parametric tests
• Look at number of groups 2 or more – t-tests (if
n<30), z-test (n>30) or ANOVA (F-test) or their
non-parametric equivalents
• For 2 or more groups check if data is paired or
independent

12. What is p-value?
Ronald Fisher

13. What is p-value?

14. What is p-value?
• The p-value is a probability of the test statistic’s sampling
distribution under the null hypothesis (null distribution, we
first assume Ho is true!)
• The (left-tailed) p-value is the quantile of the value of the
test statistic, the right-tailed p-value is one minus the
quantile, while the two-tailed p-value is twice whichever of
these is smaller.
• The p-value is NOT the probability that the null hypothesis
is true, nor is it the probability that the alternative
hypothesis is false

15. What is p-value?
• p-value is NOT the same as α !
• p-value is NOT the probability of rejecting the null
hypothesis (we reject Ho when p-value is less than
the significance level which is α)
• p-value is computed while α is set by experimental
design
• If Ho is true, α is the probability of rejecting null
hypothesis

21. CHI SQUARE OR FISHER’S EXACT TEST?
• In the days before computers were readily
available, people analyzed contingency tables by
hand, or using a calculator, using chi-square tests
• Works by computing the expected values for each
cell if the relative risk (or odds' ratio) were 1.0. It
then combines the discrepancies between
observed and expected values into a chi-square
statistic from which a P value is computed

25. CHI SQUARE OR FISHER’S EXACT TEST?
• The chi-square test is only an approximation!
• Yates continuity correction is designed to make it
better, but it over corrects so gives a p-value that
is too large (too 'conservative’)
• With large sample sizes, Yates' correction makes
little difference, and the chi-square test works very
well. With small sample sizes, chi-square is not
accurate, with or without Yates' correction

26. CHI SQUARE OR FISHER’S EXACT TEST?
• Fisher's exact test, as its name implies, always gives an
exact P value and works fine with small sample sizes
• Fisher's test (unlike chi-square) is very hard to calculate by
hand (so generally used for 2 x 2 or 2 x n table), but is easy
to compute with a computer
• Advisable to use when any cell of the table has expected
value < 5

27. CHI SQUARE OR FISHER’S EXACT TEST?
• Most statistical books advise using it instead of chi-square
test (especially small samples, but chi square becomes
acceptable for large sample sizes)
• Fisher’s exact test can be used for a m x n table
• Some have criticized it as the exact answer to the wrong
question!

28. Men
Women
Total
Dieting
a
b
a+b
Not Dieting
c
d
c+d
a+c
b+d
(a+b+c+d)=n
Total

30. ANOVA (ANALYSIS OF VARIANCE)
• The one-way analysis of variance (ANOVA) is used to
determine whether there are any significant differences
between the means of two or more independent
(unrelated) groups
• For ex: to understand if exam performance (dependent
variable) differed based on test anxiety levels amongst
students, dividing students into three independent groups
(e.g., low, medium and high-stressed students)

32. ONE-WAY ANOVA DESIGN
Treatment/C
ondition
CONDITION1
Levels (Independent Variable)
Group1
Group2
Group3
S1 DV
S2 DV
S3 DV
S4 DV
S5 DV
S6 DV
S7 DV
S8 DV
S9 DV
S10 DV
S11 DV
S12 DV
S13 DV
S14 DV
S15 DV
DV = Dependent Variable S = Subject

33. ANOVA (ANALYSIS OF VARIANCE)
• It is an omnibus test statistic and cannot tell you which
specific groups were significantly different from each
other; it only tells you that at least two groups were
different.
• Since you may have ≥3 groups in your study design,
determining which of these groups differ from each other
is done using a Post-hoc test (Tukey’s test is preferred)
which gives a Multiple comparisons table.

34. ANOVA (ANALYSIS OF VARIANCE)
• To apply ANOVA 6 assumptions must be met:
• Assumption #1: Your dependent variable should be
measured at the interval or ratio level (i.e., they are
continuous)
• Assumption #2: Your independent variable should
consist of two or more categorical, independent groups; it
can be used for just two groups (but an independentsamples t-test is more commonly used for two groups)

35. ANOVA (ANALYSIS OF VARIANCE)
• Assumption #3: You should have independence of observations,
which means that there is no relationship between the
observations in each group or between the groups themselves.
• Assumption #4: There should be no significant outliers.
• Assumption #5: Your dependent variable should be approximately
normally distributed for each category of the independent variable
(but it is quite "robust" to violations of normality)
• Assumption #6: There needs to be homogeneity of variances. (in
SPSS using Levene's test for homogeneity of variances)

36. ANOVA (ANALYSIS OF VARIANCE) METHOD
• ANOVA calculates the mean for each of the groups - the
Group Means.
• It calculates the mean for all the groups combined - the
Overall Mean.
• Then it calculates, within each group, the total deviation
of each individual's score from the Group Mean - Within
Group (Error )Variation.

37. ANOVA (ANALYSIS OF VARIANCE) METHOD
• Next, it calculates the deviation of each Group Mean
from the Overall Mean - Between Group Variation.
• Finally, ANOVA produces the F statistic which is the
ratio Between Group Variation to the Within Group
(Error) Variation.

38. ANOVA (ANALYSIS OF VARIANCE) METHOD

39. TWO-WAY ANOVA DESIGN
Treatment/Conditi
on (Independent)
Levels (Independent Variable)
Group3
S6 DV
S11 DV
S2 DV
S7 DV
S12 DV
S3 DV
S8 DV
S13 DV
S4 DV
S9 DV
S14 DV
S5 DV
S10 DV
S15 DV
S16 DV
S21 DV
S26DV
S17 DV
CONDITION2
Group2
S1 DV
CONDITION1
Group1
S22 DV
S27 DV
S18 DV
S23 DV
S28 DV
S19 DV
S24 DV
S29 DV
S20 DV
S25 DV
S30 DV

40. ANCOVA (ANALYSIS OF COVARIANCE)
• An extension of the one-way ANOVA used to determine whether
there are any significant differences between the means of two or
more independent (unrelated) groups (specifically, the adjusted
means) by adjusting for a third or confounding variable
• Third variable (known as a "covariate” or “confounding variable”) is
that you want to "statistically control” that maybe affecting results
of ANOVA
• In each one of the two groups we can compute the correlation
coefficient between the third variable and dependent variables

41. REPEATED MEASURES ANOVA
• A repeated measures ANOVA is used when you have a
single group on which you have measured something a few
times
• For example, you may have a test of understanding of
Classes. You give this test at the beginning of the topic, at
the end of the topic and then at the end of the subject
• You would use a one-way repeated measures ANOVA to see
if student performance on the test changed over time

42. REPEATED MEASURES ANOVA
• Repeated measures ANOVA is the equivalent of the one-way
ANOVA, but for related, not independent groups, and is the
extension of the dependent t-test
• A repeated measures ANOVA is also referred to as a withinsubjects ANOVA or ANOVA for correlated samples
• The major advantage with running a repeated measures ANOVA
over an independent ANOVA is that the test is generally much
more powerful. This particular advantage is achieved by the
reduction in variability (due to differences between subjects) during
the performance of the test

44. REPEATED MEASURES ANOVA
Subjects
Time/Condition (Independent Variable)
T1
T2
T3
S1
S1
S1
S1
S2
S2
S2
S2
S3
S3
S3
S3
S4
S4
S4
S4
S5
S5
S5
S5

45. TWO-WAY ANOVA REPEATED MEASURES
Factor
(Independent)
Time/Condition (Independent Variable)
Subjects
T3
S1
S1
S1
S2
S2
S2
S2
S3
S3
S3
S3
S4
S4
S4
S4
S5
S5
S5
S5
S6
S6
S6
S6
S7
GROUP2
T2
S1
GROUP1
T1
S7
S7
S7
S8
S8
S8
S8
S9
S9
S9
S9
S10
S10
S10
S10

46. Variable type & CHOOSING A Test
Explanatory
Variable
Response
Variable
Methods
Categorical
Categorical
Contingency
Tables
Categorical
Quantitative
ANOVA
Quantitative
Quantitative
Regression

47. ANOVA – WHY NOT JUST USE t-TESTS?
• Multiple t-tests are not the answer because as the number of
groups grows, the number of needed pair comparisons grows
quickly. For example in 7 groups there are 21 pairs. If we test 21
pairs we should not be surprised to observe things that happen
only 5% of the time. Thus in 21 pairings, a p-value = 0.05 for one
pair cannot be considered significant.
• Our level of significance α has to be divided for multiple
comparisons (Ex: for above it becomes α/21)
•
ANOVA puts all the data into one number (F) and gives us one pvalue for the null hypothesis.

48. ANOVA – WHY NOT JUST USE t-TESTS?
From eBook: Research skills for Psychology Majors by
William Gabrenya

49. Likert ITEM & LIKERT Scale

50. Likert ITEM & LIKERT Scale
• Likert scale consists of multiple Likert-type
items
• Likert-type scales (such as "On a scale of 1
to 10, with one being no pain and ten being
high pain, how much pain are you in today?")
• Represent ordinal data (order, rank, but
no real distance)

51. Likert ITEM & LIKERT Scale
• Fundamentally, these scales do not represent a
measurable quantity
• An individual may respond 8 and be in less pain
than someone else who responded 5
• A person may not be in exactly half as much pain if
they responded 4 than if they responded 8
• Visual Analog Scale is a Likert scale but often
(wrongly) analyzed as if it were continuous data

52. COMPOSITE SCORE & LIKERT Scale
• Composite scores combine multiple Likert item
scales into a single scale
• Composite scores must first be analyzed for
internal consistency and inter-item correlation for
each item and reported (ex: using Cronbach’s
alpha – scale reliability analysis)
• These scores represent ordinal data so must use
non-parametric tests and descriptives

53. Cronbach’s Alpha For scales
• Check for internal consistency and overall
validity of a multiple Likert-type item scale
• Check correlation (α) with each item
deleted at a time
• Based on number of items and comparison
of its variances

54. Cronbach’s Alpha For scales
• Values of α range from 0 to 1
• Ideally overall α and α for each item (when
deleted from scale) must be > 0.7 to 0.8
• Clinical scores need higher α > 0.8 to 0.9
(Bland-Altman)

55. Power analysis & effect size
• To calculate sample size (n) we must know the type
of statistical test involved in our primary outcome
measure
• Also we must also know:
• Desired α error (usually taken as 0.05)
• Power (1-β) usually taken 0.8 (80%) or greater
• Two or one-tailed comparison
• Effect size

56. Power analysis & effect size
• Power is the fraction of experiments that you expect to
yield a "statistically significant” p-value (80% of
experiments of the sample may yield a significant p-value)
• Effect size (Cohen’s d for mean) depends on study design,
it is calculated by data from pilot studies or reference
studies
• Effect size depends on a clinically defined level of
significance (ex: more than 20% difference between 2
groups, with difference for proportion or mean ± SD data
etc)

57. Power analysis & effect size
• Cohen’s d is usually calculated based on pilot studies but if
effect size is unknown Jacob Cohen provided 3 guess estimate
effect sizes (value varies slightly for different statistical tests):
1. Small effect d around 0.2 (requires large sample sizes)
2. Medium effect d around 0.5 (seen with careful observation, use
when in doubt)
3. Large effect greater than 0.8 (if large it is obvious)
• Criticized when d is used as above as “T-shirt” effect sizes

58. Power analysis & effect size
• Calculation of required sample size a with set target for
power before starting the final study is called A priori
analysis (before the fact) – accepted method, especially
important to avoid incorrectly being “blind” to a real
difference in a negative study (due to large βerror)
• Calculation of required sample size at the end of the final
study is called Post hoc analysis (after the fact) – incorrect
as the computed power is a simple reflection of the pvalue!
• G*Power software is a free useful resource