1. Introduction to Biostatistics
Georgi Iskrov, PhD
Department of Social Medicine

2. Before we start

3. Outline Population vs sample Descriptive vs inferential statistics
Sampling methods
Sample size calculation
Level of measurement
Graphical summaries

4. Outline Descriptive statistics Measures of central tendency
Measures of spread
Normal distribution
Central limit theorem
Outliers
Inferential statistic
Confidence interval

5. Definition of biostatistics
The science of
collecting, organizing, analyzing, interpreting and presenting data
for the purpose of
more effective decisions in clinical context.

6. Why do we need to use statistical methods?
To make strongest possible conclusion from limited amounts of data;
To generalize from a particular set of data to a more general conclusion.
What do we need to pay attention to?
Bias
Probability
Statistics means
never having to say you are certain!

7. Population vs Sample Population Parameters μ, σ, σ2
Sample / Statistics
x, s, s2

8. Population vs Sample
Population includes all objects of interest whereas sample is only a portion of the population.
Parameters are associated with populations and statistics with samples
Parameters are usually denoted using Greek letters (μ, σ) while statistics are usually denoted using Roman letters (X, s)
There are several reasons why we do not work with populations.
They are usually large, and it is often impossible to get data for every object we're studying
Sampling does not usually occur without cost, and the more items surveyed, the larger the cost

9. Descriptive vs Inferential statistics
We compute statistics, and use them to estimate parameters.
The computation is the first part of the statistical analysis (Descriptive Statistics) and the estimation is the second part (Inferential Statistics).
Descriptive Statistics
The procedure used to organize and summarize masses of data
Inferential Statistics
The methods used to find out something about a population, based on a sample

10. Descriptive vs Inferential statistics
Population
Parameters
Sampling
From population to sample
Sample
Statistics
From sample to population
Inferential statistics

11. Sampling
When a sample is drawn, there is no certainty that it will be representative for the population.
Sample A
Sample B

12. Error Random error can be conceptualized as sampling variability.
Bias (systematic error) is a difference between an observed value and the true value due to all causes other than sampling variability.
Accuracy is a general term denoting the absence of error of all kinds.

13. Sampling
Sampling
A specific principle used to select members of population to be included in the study.
Due to the large size of target population, researchers have no choice but to study the a number of cases of elements within the population to represent the population and to reach conclusions about the population.
Biased sample
Biased sample is one in which the method used to create the sample results in samples that are systematically different from the population.
Random sample
In random sampling, each item or element of the population has an equal chance of being chosen at each draw.

14. Sampling
Sample B
Sample A
Population

15. Sampling
Sample B
Sample A
Population

16. Sampling Stages of sampling: Defining target population
Determining sampling size
Selecting a sampling method
Properties of a good sample:
Random selection
Representativeness by structure
Representativeness by number of cases

17. Sampling
Random sampling: Sample group members are selected in a random manner
Highly effective if all subjects participate in data collection
High level of sampling error when sample size is small
Systematic: Including every Nth member of population in the study
Time efficient
Cost efficient
High sampling bias if periodicity exists

18. Sampling
Judgement: Sample group members are selected on the basis of judgement of researcher
Time efficiency
Samples are not highly representative
Unscientific approach
Personal bias
Convenience: Obtaining participants conveniently with no requirements whatsoever
High levels of simplicity and ease
Usefulness in pilot studies
Highest level of sampling error
Selection bias

19. Sampling
Snowball: Sample group members nominate additional members to participate in the study
Possibility to recruit hidden population
Over-representation of a particular network
Reluctance of sample group members to nominate additional members

20. Sampling Stratified: Representation of specific subgroup or strata
Effective representation of all subgroups
Precise estimates in cases of homogeneity or heterogeneity within strata
Knowledge of strata membership is required
Complex to apply in practical levels
Cluster: Clusters of participants representing population are identified as sample group members
Time and cost efficient
Group-level information needs to be known
Usually higher sampling errors compared to alternative sampling methods

21. Sample size calculation
Law of Large Numbers: As the number of trials of a random process increases, the percentage difference between the expected and actual values goes to zero.
Application in biostatistics: Bigger sample size, smaller margin of error.
A properly designed study will include a justification for the number of experimental units (people/animals) being examined.
Sample size calculations are necessary to design experiments that are large enough to produce useful information and small enough to be practical.

22. Sample size calculation
Generally, the sample size for any study depends on:
Acceptable level of confidence;
Expected effect size and absolute error of precision;
Underlying scatter in the population;
Power of the study.
High power
Large sample size
Large effect
Little scatter
Low power
Small sample size
Small effect
Lots of scatter

23. Sample size calculation
For quantitative variables:
Z – confidence level;
SD – standard deviation;
d – absolute error of precision.

24. Sample size calculation
For quantitative variables:
A researcher is interested in knowing the average systolic blood pressure in pediatric age group at 95% level of confidence and precision of 5 mmHg. Standard deviation, based on previous studies, is 25 mmHg.
=> 97

25. Sample size calculation
For qualitative variables:
Z – confidence level
p – expected proportion in population
d – absolute error of precision

26. Sample size calculation
For qualitative variables:
A researcher is interested in knowing the proportion of diabetes patients having hypertension. According to a previous study, the actual number is no more than 15%. The researcher wants to calculate this size with a 5% absolute precision error and a 95% confidence level.
=> 196

27. When do you need biostatistics?
BEFORE you start your study!
After that, it will be too late…

28. Planning Research programme: Aim Object Units of observation
Indices of observation
Place
Time
Statistical analyses
Methodology

29. Planning
Aim
The aim of the investigation is trying to summarize and formulate clearly the research hypothesis.
Object
Object of the investigation is the event, that is going to be studied.
Units of observation
Logical unit – each studied case
Technical unit – the environment, where the logical units are situated
Indices of observation – not too many, but important; measurable; additive and self controlling.
Factorial
Resultative

30. Planning
Place
Time
Single – events are studied in a single moment of time, the so called “critical moment”.
Continuous – used to characterize a long term tendency of the events
Statistical analyses
Methodology

31. Quantitative (metric) variables
Continuous
Measured units
Metric continuous variables can be properly measured and have units of measurement.
Continuous values on proper numeric line or scale
Data are real numbers (located on the number line).
Discrete
Integer values on proper numeric line or scale
Metric discrete variables can be properly counted and have units of measurement – ‘numbers of things’.
Counted units

32. Qualitative (categorical) variables
Nominal
Values in arbitrary categories
Ordering of the categories is completely arbitrary. In other words, categories cannot be ordered in any meaningful way.
No units!
Data do not have any units of measurement.
Ordinal
Values in ordered categories
Ordering of the categories is not arbitrary. It is now possible to order the categories in a meaningful way.

33. Levels of measurement
There are four levels of measurement: Nominal, Ordinal, Interval, and Ratio. These go from lowest level to highest level.
Data is classified according to the highest level which it fits. Each additional level adds something the previous level didn't have.
Nominal is the lowest level. Only names are meaningful here.
Ordinal adds an order to the names.
Interval adds meaningful differences.
Ratio adds a zero so that ratios are meaningful.

34. Levels of measurement Nominal scale – eg., genotype
You can code it with numbers, but the order is arbitrary and any calculations would be meaningless.
Ordinal scale – eg., pain score from 1 to 10
The order matters but not the difference between values.
Interval scale – eg., temperature in C
The difference between two values is meaningful.
Ratio scale – eg., height
It has a clear definition of 0. When the variable equals 0, there is none of that variable. When working with ratio variables, but not interval variables, you can look at the ratio of two measurements.

35. Descriptive statistics
Organising data
Tables
Frequency distributions
Relative frequency distributions
Graphs
Bar chart
Histogram
Box plot
Summarising data
Central tendency (location)
Variation (spread)

36. Graphical summaries Variable Graph Statistics One qualitative
Bar chart
Pie chart
Frequency table
Relative frequency table
Proportion
Two qualitative
Side-by-side bar chart
Segmented bar chart
Two-way table
Difference in proportions
One quantitative
Dotplot
Histogram
Boxplot
Measures of central tendency
Measures of spread
Other: five number summary, percentiles, distribution shape
One quantitative by one qualitative
Side-by-side boxplots
Stacked dotplots
Statistics broken down by group
Difference in means
Two quantitative
Scatterplot
Correlation

37. Frequency distribution
Frequency distribution of survival for both groups
Survival Frequency
14 2
17 1
21 1
22 2
23 1
24 2
25 1
27 1
28 1
29 1
31 1
33 1
34 2
35 1
39 1
41 1
Total 20
Experimental group (10 patients)
Individual survival in months:
23 27
17 34
41 28
22 33
29 14
Classes of values
Control group (10 patients)
Individual survival in months:
24 31
39 35
34 24
14 21
25 22

38. Relative frequency distribution
Relative frequency distribution of survival for both groups
Survival Frequency Percent Cumulative percent
% 10%
% 15%
% 20%
% 30%
% 35%
% 45%
% 50%
% 55%
% 60%
% 65%
% 70%
% 75%
% 85%
% 90%
% 95%
% 100%
Total % 38

39. Grouped relative frequency distribution
Relative frequency distribution of survival for both groups
Survival Frequency Percent Cumulative Percent
10 – % 10%
15 – % 15%
20 – % 45%
25 – % 65%
30 – % 85%
35 – % 95%
40 – % 100%
Total %
Classes of intervals
What rules to follow when groupping data?

40. Descriptive statistics
Summarising data:
Central tendency (or sample’s middle value)
Mean
Median
Mode
Spread (or summary of differences within groups)
Range
Interquartile range
Variance
Standard deviation

41. Mean Most commonly called average. Experimental group (10 patients)
Individual survival in months:
23 27
17 34
41 28
22 33
29 14
Experimental group (10 patients)
Individual survival in months:
23 27
17 34
41 28
22 33
29 14
Control group (10 patients)
Individual survival in months:
14 21
25 22

42. Median The middle value when a variable’s values are ranked in order.
The point that divides a distribution into two equal halves.
When data are listed in order, the median is the point at which 50% of the cases are above and 50% below it.
The 50th percentile.

43. Median Control group (10 patients) Individual survival in months: 1421 22 24 25 31 34 35 39
Median = 24.5
(five cases above, five below)

44. Median
If the recorded values for a variable form a symmetric distribution, the median and mean are identical.
In skewed data, the mean lies further toward the skew than the median.
Symmetric
Skewed
Mean
Mean
Median
Median

45. Mode The most common data point is called the mode.
Individual survival data for the control group are:
14, 21, 22, 24, 24, 25, 31, 34, 35, 39
It is possible to have more than one mode.
If all values are unique, there is no mode.
Mode may mot be at the center of a distribution.

46. Mode
It may give you the most likely experience rather than the typical or central experience.
In symmetric distributions, the mean, median and mode are the same.
In skewed data, the mean and median lie further toward the skew than the mode.
Skewed
Symmetric
Mean
Median
Mode
Mode
Median
Mean

47. Spread Variation of the recorded values on a variable.
The larger the spread, the further the individual cases are from the mean.
The smaller the spread, the closer the individual scores are to the mean.
Mean
Mean

48. Range
The spread, or the distance, between the lowest and highest values of a variable.
To get the range for a variable, you subtract its lowest value from its highest value.
Experimental group (10 patients)
Individual survival in months:
23 27
17 34
41 28
22 33
29 14
Range = 41 – 14 = 27
Control group (10 patients)
Individual survival in months:
24 31
39 35
34 24
14 21
25 22
Range = 39 – 14 = 25

49. Standard deviation
Standard deviation takes into account all individual deviations.
A deviation is the distance away from the mean of a case’s score.
Experimental group’s SD = 8.13 months
Control group’s SD = 7.64 months

50. Standard deviation
The larger standard deviation, the greater amounts of variation around the mean.
Standard deviation is equal to 0, only when all values are the same.
Like the mean, the standard deviation will be inflated by an outlier case value.

51. Interquartile range
The interquartile range (IQR) is a measure of variability, based on dividing a data set into quartiles.
Quartiles divide a rank-ordered data set into four equal parts. The values that divide each part are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.
IQR is equal to Q3 minus Q1.

52. Central tendency and spread
Central tendency: Mean, mode and median
Spread: Range, interquartile range, standard deviation
Mistakes:
Focusing on only the mean and ignoring the variability
Standard deviation and standard error of the mean
Variation and variance
What is best to use in different scenarios?
Symmetrical data: mean and standard deviation
Skewed data: median and interquartile range

53. Boxplot

54. Important rules
When a constant is added to every observation, the new sample mean is equal to original mean plus the constant.
When a constant is added to every observation, the standard deviation is unaffected.
When every observation is multiplied by the same constant, the new sample mean is equal to original mean multiplied by the constant.
When every observation is multiplied by the same constant, the new sample standard deviation is equal to original standard deviation multiplied by the magnitude of the constant.

55. Normal (Gaussian) distribution
Mean and standard deviations are a particularly appropriate summary for data whose histogram approximates a normal distribution (the bell-shaped curve).
If you say that a set of data has a mean survival of 29 months, the typical listener will picture a bell-shaped curve centered with its peak at 29 months.

56. Rule of 3-sigma When data are approximately normally distributed:
approximately 68% of the data lie within one SD of the mean;
approximately 95% of the data lie within two SDs of the mean;
approximately 99% of the data lie within three SDs of the mean.

57. Normal (Gaussian) distribution
Central limit theorem:
Create a population with a known distribution that is not normal;
Randomly select many samples of equal size from that population;
Tabulate the means of these samples and graph the frequency distribution.
Central limit theorem states that if your samples are large enough, the distribution of the means will approximate a normal distribution even if the population is not Gaussian.
Mistakes:
Normal vs common (or disease free);
Few biological distributions are exactly normal.

58. Outliers
Values that lie very far away from the other values in the data set.

59. Outliers Outliers can occur for several reasons: Mistakes:
Invalid data entry
Biological diversity
Random chance
Experimental error
Skewed distribution
Mistakes:
Not realizing that outliers are common in data sampled from skewed distribution
Eliminating outliers only when you do not get the results you want
Truly removing outliers from your records

60. Outliers Outlier test:
If values are sampled from a normal distribution, what is the chance one value will be as far from the others as the extreme value observed?
Examples: Chauvenet criterion, Grubbs test, Peirce criterion
Nevertheless, deletion of outlier data is generally a controversial practice!

61. Inferential statistics
Population
Parameters
Sampling
From population to sample
Sample
Statistics
From sample to population
Inferential statistics

62. Confidence interval for the population mean
Population mean: point estimate vs interval estimate
Standard error of the mean – how close the sample mean is likely to be to the population mean.
Assumptions: a random representative sample, independent observations, the population is normally distributed (at least approximately).
Confidence interval depends on: sample mean, standard deviation, sample size, degree of confidence.
Mistakes:
95% of the values lie within the 95% CI;
A 95% CI covers the mean ± 2 SD.

63. Standard error of mean
The sample mean estimates individual values. The uncertainty with which this mean estimates individual values is given by the standard deviation.
The sample mean estimates the population mean. The uncertainty with which this mean estimates the population mean is given by the standard error of the mean.

64. Confidence interval for the population mean
The confidence interval for the mean gives us a range of values around the mean where we expect the “true” population mean is located.
95% confidence interval for the population mean is:

65. Confidence interval for the population mean
The duration of time from first exposure to HIV infection to AIDS diagnosis is called the incubation period. The incubation periods (in years) of a random sample of 30 HIV infected individuals are: 12.0, , 9.5, 6.3, 13.5, 12.5, 7.2, 12.0, 10.5, 5.2, 9.5, 6.3, 13.1, 13.5, , 10.7, 7.2, 14.9, 6.5, 8.1, 7.9, 12.0, 6.3, 7.8, 6.3, 12.5, 5.2, 13.1, , 7.2. Calculate the 95% CI for the population mean incubation period in HIV.
X = 9.5 years; SD = 2.8 years
SEM = 0.5 years
95% level of confidence => Z = 1.96
µ = 9.5 ± (1.96 x 0.5) = 9.5 ± 1 years
95% CI for µ is (8.5; 10.5 years)

66. Confidence interval for the population mean
X = 9.5 years; SD = 2.8 years
SEM = 0.5 years
95% level of confidence => Z = 1.96
µ = 9.5 ± (1.96 x 0.5) = 9.5 ± 1 years
95% CI for µ is (8.5; 10.5 years)
99% level of confidence => Z = 2.58
µ = 9.5 ± (2.58 x 0.5) = 9.5 ± 1.3 years
99% CI for µ is (8.2; 10.8 years)

67. Describing qualitative data
Improvement
No improvement
Total
Gluten-free diet 54 46
100
No gluten-free diet 47 53
101 99
200

68. Describing qualitative data
Standard error of proportion:
The 95% confidence interval for a population proportion is: