1. Descriptive and inferential statistics
Asst. Prof. Georgi Iskrov, PhD
Department of Social Medicine

2. Lecture slides to be updated!
Before we start
Lecture slides to be updated!

3. Outline Statistics Sample, population and sampling
Descriptive and inferential statistics
Types of variables and level of measurement
Measures of central tendency and spread
Normal distribution
Confidence intervals
Sample size calculation
Hypothesis testing
Significance, power and errors
Normality tests

4. 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

5. Population vs Sample Population Parameters μ, σ Sample / Statistics
x, s

6. 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 are studying;
Sampling does not usually occur without cost. The more items surveyed, the larger the cost.

7. Inferential statistics
Population
Parameters
Sampling
From population to sample
Sample
Statistics
From sample to population
Inferential statistics 7

8. 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.

9. Sampling
Individuals in the population vary from one another with respect to an outcome of interest.

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

11. Sampling
Sample B
Sample A
Population

12. Sampling
Sample B
Sample A
Population

13. Sampling
Random sample: In random sampling, each item or element of the population has an equal chance of being chosen at each draw. While this is the preferred way of sampling, it is often difficult to do. It requires that a complete list of every element in the population be obtained. Computer generated lists are often used with random sampling.
Properties of a good sample:
Random selection;
Representativeness by structure;
Representativeness by number of cases.

14. Sampling
Systematic sampling: The list of elements is “counted off”. That is, every k-th element is taken. This is similar to lining everyone up and numbering off “1,2,3,4; 1,2,3,4; etc”. When done numbering, all people numbered 4 would be used.
Convenience sampling: In convenience sampling, readily available data is used. That is, the first people the surveyor runs into.

15. Sampling
Cluster sampling: It is accomplished by dividing the population into groups (clusters), usually geographically. The clusters are randomly selected, and each element in the selected clusters are used.
Stratified sampling: It divides the population into groups, called strata. However, this time it is by some characteristic, not geographically. For instance, the population might be separated into males and females. A sample is taken from each of these strata using either random, systematic, or convenience sampling.

16. Random and systematic errors
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.
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.
Accuracy is a general term denoting the absence of error of all kinds.

17. 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.

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

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

20. 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.

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

22. 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.

23. Frequency distribution Mean, standard deviation
Variables
Different types of data require different kind of analyses.
Nominal
Ordinal
Interval
Ratio
Frequency distribution
Yes
Median, percentiles No
Mean, standard deviation

24. 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 did not 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.

25. 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.

26. 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

27. Normal (Gaussian) distribution
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 99.7% of the data lie within three SDs of the mean.

28. 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.

29. 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.

30. 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)

31. 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)

32. Is there a difference? Hypothesis testing Diabetes type 2 study
Experimental group: Mean blood sugar level: 103 mg/dl
Control group: Mean blood sugar level: 107 mg/dl
Pancreatic cancer study
Experimental group: 1-year survival rate: 23%
Control group: 1-year survival rate: 20%
Is there a difference?

33. Hypothesis testing The general idea of hypothesis testing involves:
Making an initial assumption;
Collecting evidence (data);
Based on the available evidence (data), deciding whether to reject or not reject the initial assumption.
Every hypothesis test – regardless of the population parameter involved – requires the above three steps.

34. Null hypothesis – H0 This is the hypothesis under test, denoted as H0.
The null hypothesis is usually stated as the absence of a difference or an effect;
The null hypothesis says there is no effect;
The null hypothesis is rejected if the significance test shows the data are inconsistent with the null hypothesis.

35. Alternative hypothesis – H1
This is the alternative to the null hypothesis. It is denoted as H', H1, or HA.
It is usually the complement of the null hypothesis;
If, for example, the null hypothesis says two population means are equal, the alternative says the means are unequal.

36. Criminal trial
Criminal justice system assumes “the defendant is innocent until proven guilty”. That is, our initial assumption is that the defendant is innocent.
In the practice of statistics, we make our initial assumption when we state our two competing hypotheses – the null hypothesis (H0) and the alternative hypothesis (HA). Here, our hypotheses are:
H0: Defendant is not guilty (innocent);
HA: Defendant is guilty;
In statistics, we always assume the null hypothesis is true. That is, the null hypothesis is always our initial assumption.

37. Criminal trial
The prosecution team then collects evidence with the hopes of finding “sufficient evidence” to make the assumption of innocence refutable.
In statistics, the data are the evidence.
The jury then makes a decision based on the available evidence:
If the jury finds sufficient evidence – beyond a reasonable doubt – to make the assumption of innocence refutable, the jury rejects H0 and deems the defendant guilty. We behave as if the defendant is guilty.
If there is insufficient evidence, then the jury does not reject H0. We behave as if the defendant is innocent.

38. Making the decision
Recall that it is either likely or unlikely that we would observe the evidence we did given our initial assumption.
If it is likely, we do not reject the null hypothesis;
If it is unlikely, then we reject the null hypothesis in favor of the alternative hypothesis;
Effectively, then, making the decision reduces to determining “likely” or “unlikely”.

39. Making the decision
In statistics, there are two ways to determine whether the evidence is likely or unlikely given the initial assumption:
We could take the “critical value approach” (favored in many of the older textbooks).
Or, we could take the “p-value approach” (what is used most often in research, journal articles, and statistical software).

40. Making the decision
Suppose we find a difference between two groups in survival:
patients on a new drug have a survival of 15 months;
patients on the old drug have a survival of 18 months.
So, the difference is 3 months.

41. Making the decision
Suppose we find a difference between two groups in survival:
patients on a new drug have a survival of 15 months;
patients on the old drug have a survival of 18 months.
So, the difference is 3 months.
Do we accept or reject the hypothesis of no true difference between the groups (the two drugs)?
Is a difference of 3 a lot, statistically speaking – a huge difference that is rarely seen?
Or is it not much – the sort of thing that happens all the time?

42. Making the decision
A statistical test tells you how often you would get a difference of 3, simply by chance, if the null hypothesis is correct – no real difference between the two groups.
Suppose the test is done and its result is that p = This means that you’d get a difference of 3 quite often just by the play of chance – 32 times in 100 – even when there is in reality no true difference between the groups.

43. Making the decision
A statistical test tells you how often you’d get a difference of 3, simply by chance, if the null hypothesis is correct – no real difference between the two groups.
On the other hand if we did the statistical analysis and p = , then we say that you would only get a difference as big as 3 by the play of chance 1 time in That is so rarely that we want to reject our hypothesis of no difference: there is something different about the new therapy.

44. Hypothesis testing
Somewhere between 0.32 and we may not be sure whether to reject the null hypothesis or not.
Mostly we reject the null hypothesis when, if the null hypothesis were true, the result we got would have happened less than 5 times in 100 by chance. This is the ‘conventional’ cutoff of 5% or p <0.05.
This cutoff is commonly used but it is arbitrary i.e. no particular reason why we use 0.05 rather than 0.06 or or whatever.

45. Hypothesis testing Decision: Reject null hypothesis
Do not reject null hypothesis
Null hypothesis is true
Type I error
No error
Null hypothesis is false
Type II error

46. Type I and II errors
A type I error is the incorrect rejection of a true null hypothesis (also known as a “false positive” finding).
The probability of a type I error is denoted by the Greek letter  (alpha).
A type II error is incorrectly retaining a false null hypothesis (also known as a "false negative" finding).
The probability of a type II error is denoted by the Greek letter  (beta).

47. Level of significance
Level of significance (α) – the threshold for declaring if a result is significant. If the null hypothesis is true, α is the probability of rejecting the null hypothesis.
α is decided as part of the research design, while p-value is computed from data.
α = 0.05 is most commonly used.
Small α value reduces the chance of Type I error, but increases the chance of Type II error.
Trade-off based on the consequences of Type I (false-positive) and Type II (false-negative) errors.

48. Power
Power – the probability of rejecting a false null hypothesis. Statistical power is inversely related to β or the probability of making a Type II error (power is equal to 1 – β).
Power depends on the sample size, variability, significance level and hypothetical effect size.
You need a larger sample when you are looking for a small effect and when the standard deviation is large.

49. Choosing a statistical test
Choice of a statistical test depends on:
Level of measurement for the dependent and independent variables
Number of groups or dependent measures
Number of units of observation
Type of distribution
The population parameter of interest (mean, variance, differences between means and/or variances)

50. Choosing a statistical test
Multiple comparison – two or more data sets, which should be analyzed
repeated measurements made on the same individuals;
entirely independent samples.
Degrees of freedom – the number of scores, items, or other units in the data set, which are free to vary
One- and two tailed tests
one-tailed test of significance used for directional hypothesis;
two-tailed tests in all other situations.
Sample size – number of cases, on which data have been obtained
Which of the basic characteristics of a distribution are more sensitive to the sample size?

51. Student t-test 51

52. 2-sample t-test Aim: Compare two means
Example: Comparing pulse rate in people taking two different drugs
Assumption: Both data sets are sampled from Gaussian distributions with the same population standard deviation
Effect size: Difference between two means
Null hypothesis: The two population means are identical
Meaning of P value: If the two population means are identical, what is the chance of observing such a difference (or a bigger one) between means by chance alone?

53. Paired t-test
Aim: Compare a continuous variable before and after an intervention
Example: Comparing pulse rate before and after taking a drug
Assumption: The population of paired differences is Gaussian
Effect size: Mean of the paired differences
Null hypothesis: The population mean of paired differences is zero
Meaning of P value: If there is no difference in the population, what is the chance of observing such a difference (or a bigger one) between means by chance alone?

54. One-way ANOVA Aim: Compare three or more means
Example: Comparing pulse rate in 3 groups of people, each group taking a different drug
Assumption: All data sets are sampled from Gaussian distributions with the same population standard deviation
Effect size: Fraction of the total variation explained by variation among group means
Null hypothesis: All population means are identical
Meaning of P value: If the population means are identical, what is the chance of observing such a difference (or a bigger one) between means by chance alone?

55. Parametric and non-parametric tests
Parametric test – the variable we have measured in the sample is normally distributed in the population to which we plan to generalize our findings
Non-parametric test – distribution free, no assumption about the distribution of the variable in the population

56. Normality test
Normality tests are used to determine if a data set is modeled by a normal distribution and to compute how likely it is for a random variable underlying the data set to be normally distributed.
In descriptive statistics terms, a normality test measures a goodness of fit of a normal model to the data – if the fit is poor then the data are not well modeled in that respect by a normal distribution, without making a judgment on any underlying variable.
In frequentist statistics statistical hypothesis testing, data are tested against the null hypothesis that it is normally distributed.

57. Normality test Graphical methods
An informal approach to testing normality is to compare a histogram of the sample data to a normal probability curve. The empirical distribution of the data (the histogram) should be bell-shaped and resemble the normal distribution. This might be difficult to see if the sample is small.

58. Normality test Frequentist tests
Tests of univariate normality include the following:
D'Agostino's K-squared test
Jarque–Bera test
Anderson–Darling test
Cramér–von Mises criterion
Lilliefors test
Kolmogorov–Smirnov test
Shapiro–Wilk test
Etc.

59. Normality test Kolmogorov–Smirnov test
K–S test is a nonparametric test of the equality of distributions that can be used to compare a sample with a reference distribution (1-sample K–S test), or to compare two samples (2-sample K–S test).
K–S statistic quantifies a distance between the empirical distribution of the sample and the cumulative distribution of the reference distribution, or between the empirical distributions of two samples.
The null hypothesis is that the sample is drawn from the reference distribution (in the 1-sample case) or that the samples are drawn from the same distribution (in the 2-sample case).

60. Normality test Kolmogorov–Smirnov test
In the special case of testing for normality of the distribution, samples are standardized and compared with a standard normal distribution. This is equivalent to setting the mean and variance of the reference distribution equal to the sample estimates, and it is known that using these to define the specific reference distribution changes the null distribution of the test statistic.