Measures of Disease Frequency

• Rates• Ratios• Proportions

Ratio

• The most basic measure of distribution. • Expresses a relation in the size between two

random quantities.• Numerator is not a component of denominator.• X : Y = X/Y (simply dividing one quantity by another)

Numerator and Denominator are mutually exclusive.

•Example: The number of stillbirths per thousand live births.

Proportion• A proportion is a ratio which indicates the

relation in magnitude of a part to the whole.• Numerator is always included in the

denominator.• Usually expressed as percentage.• For example: the proportion of women over

the age of 50 who have had a hysterectomy, or the number of fetal deaths out of the total number of births (live births plus fetal deaths).

Rate• 500 deaths from motor vehicle accidents in city A

during 1985.• For comparison between City A and City B

calculate Rate.• Rate measures the occurrence of some particular

event in a population during a given time period.• For example, the number of newly diagnosed

cases of breast cancer per 100,000 women during a given year.

A Proportion with specifications of Time

• Distinct Relationship between Numerator & Denominator.

• Death Rate: CDR, IMR, MMR.• Specific Rate: Disease specific, age – group

specific, specific time periods.• Standardized rates: By direct method and

indirect method.

Incidence“The number of new cases occurring in a defined

population during a specified period of time”

Number of new cases of specific disease during a given time period

= -----------------------------------------------------X 1000Population at risk during that period

7

Issues in the Calculation of Incidence

For any measure of disease frequency, precise definition of the denominator is essential for both accuracy and clarity. This is a particular concern in the calculation of incidence. The denominator of a measure of incidence should include only those who are considered "at risk" of developing the disease.

8

Special Types of Incidence RatesMorbidity Rate: the incidence rate of non fatal cases in the total population at risk during a specified period of time, e.g., the morbidity rate of TB in the US in 1982 was calculated by dividing the number of nonfatal cases newly reported during that year by the total US midyear population.

Total no of nonfatal cases of TB = 25,520/ 231,534,000Mid year Population

= 11.0 per 100,000 population

• Expresses the incidence of deaths in a particular population during a specific period of time.

• Calculated by dividing the number of fatalities during that period by the total population.

Mortality Rate

Prevalence• All current cases (old & new) existing at a

given point of time, or over a period of time in a given population.

• Definition: the total number of all individuals who have an attribute or disease at a particular time divided by the population at risk of having the attribute or disease at this point in time or midway through the period

Point prevalence

• The number of all current (old and new) cases of a disease at one point in time in relation to a defined population.

Number of all current (old &new) cases of specific disease existing at a given point in time = ----------------------------------------------------- X 1000

Estimated Population at the same point of time

Period prevalence

• The number of all current (old and new) cases existing during a defined period of time expressed in relation to a defined population.

Number of existing cases (old &new) of a specified disease during a given period of time interval

= ----------------------------------------------------- X 1000 Estimated mid-interval Population at risk

Measures of Central Tendency

• Mean

• Median

• Mode

• If the whole sheets of information are presented to anybody he/she cannot make any meaning out of it and if the question is such that you should present the gist of information in a single word, then it becomes more difficult.

• History tells us that this question was answered by giving the average value of the data.

• Take the example of the following data of ages of the children:

2,3,4,4,4,5,6 -------- (Data I)

Mean

Also known as the AVERAGE. It Is calculated by totaling the results of all the observations and dividing by the total number of observations. Note that the mean can only be calculated for numerical data.

Mode

• Scientists after a lot of discussion gave

solution to the problem by recommending

that the most frequently occurring value (most

repeated) should be taken as representative

because it will involve most of the people and

thus most of the people will be benefited and

it was termed as MODE. (TB, Malaria, Typhoid)

Mode Examples

• No Mode Raw Data:10.3 4.9 8.9 11.7 6.3 7.7

• One Mode Raw Data: 2 3 4 4 4 5 6

• More Than 1 Mode Raw Data: 21 28 28 41 43 43

Median

• Take the example No. l in the data I

2,3,4,4,4,5,6 • If we divide the data in two equal groups after

arranging the data in ascending or descending order, it will appear like this

• Serial No. 1,2,3, 4, 5,6,7

• Value 2,3,4, 4, 4,5,6

Median

Measure of central tendencyMiddlemost or most central item in the

set of ordered numbers– If odd n, middle value of sequence– If even n, average of 2 middle values

Not affected by extreme values

Calculating Median

Median = item in the data array( )thn + 1 2

Number of items in the array

Median of Odd Sample Size

Median

Positioning Point

Median

n 12

7 12

4 0

4

.

Number 1 2 3 4 5 6 7of Child

Age 2 3 4 4 4 5 6

Ages of the children

Median of Even Sample Size

Median is 4

Positioning Point

Median

n 12

8 12 4.5

4 42

4

Ages of the children

Number of Child 1 2 3 4 5 6 7 8

Age 2 3 4 4 4 5 6 60

Data Analysis & Statistics

• Measures of central tendency: mean, median, mode– Mean: average– Median: exact middle number– Mode: most frequently occurring number

Example:67, 94, 72, 88, 88, 95, 100, 88, 72Mean = 84.9, Medium = 88, Mode = 88

Range is defined as the difference in value between the highest (maximum) and the lowest (minimum) observation e.g. for previous data the lowest value is 2 and highest is 6

Hence the range is 6-2 = 4

Measures of Variation

Cont…

Standard Deviation• A measure, which describes how much

individual measurements differ, on the average, from the mean.

• A large standard deviation shows that there is a wide scatter of measured values around the mean, while a small standard deviation shows that the individual values are concentrated around the mean with little variation among them.

Data analysis

• Central tendency numbers are affected by extreme scores, therefore, measures of variation must also be considered: range and standard deviation– A large standard deviation means that

scores are very spread out from the mean.– A small standard deviation means that

scores are relatively close to the mean.

27

Standard error of the mean• When we draw a sample from study population and

compute its sample mean it is not likely to be identical to the population mean. If we draw another sample from same population and compute its sample mean, this may also not be identical to the first sample mean. It probably also differs from the true mean of the total population from which the sample was drawn this phenomenon is called sampling variation.

• Standard error: gives an estimate of the degree to which the sample mean varies from the population mean and this measures is used to calculate Confidence Interval.

28

The Normal Distribution

• Many variables have a normal distribution. This is a bell shaped curve with most of the values clustered near the mean and a few values out near the tails.

29

• The normal distribution is symmetrical around the mean. The mean, the median and the mode of a normal distribution have the same value.

• An important characteristic of a normally distributed variable is that 95% of the measurements have value which are approximately within 2 standard deviations (SD) of the mean.

30

The Normal Distribution

Characteristics of Normal Distribution (Gaussian)

• The normal distribution curve is bell shaped.• The mean, median and mode are located at

the centre of the distribution.• The normal distribution curve is unimodal (i.e.

it has only one mode).• The curve is symmetrical about the mean, that

is the shape is same on both sides of a vertical line passing through the centre.

31

Characteristics of Normal Distribution

• The curve is continuous, i.e. there are no gaps or holes. For each value of ‘x’ there is a corresponding value of ‘y’.

• The curve never touches the ‘x’ axis. With extension – gets increasingly closer.

• The total area under the curve is 100%. • Called “Normal” because..........

32

Characteristics of Normal Distribution

• The area under the curve:– With in ±1 Standard Deviation of mean is

approx 68% (outside is 32%)– With in ±2 Standard Deviation of mean is

approx 95% (outside is 5%)– With in ±3 Standard Deviation of mean is

approx 99.7% (outside is 0.3%)• Do remember the exact figures for these three

parameters. (68.3, 95.5, 99.7)

33

34

Factors Affecting Study Outcomes

What is Bias?

• Defined as an error in sampling or testing that systematically under- or over-represents one outcome (answer) over the other.

• It is the exact opposite of randomness.• All nonrandom sampling techniques are open to

some sort of bias.• Presence of bias in a study will make inferences

less meaningful.

Types of Biases

There are basically only three major types of biases.

Selection bias Response bias Information bias

All others are simply varieties of these three types

Selection Bias

– Caused by nonrandom sampling, so that a systematic difference is present between people selected for the study and people not selected for the study.

– Can be caused by convenience sampling, patient referral patterns, survival differences or loss to follow-up.

– An avoidable bias, if not eliminated, can ruin the chances of acceptance or publication of the study.

Response Bias– a (selection) bias where respondents differ

systematically from non-respondents.– For example, people with a certain disease (or

their relatives) are more likely to respond to oral or written questions than normal people.

– More educated people are more likely to provide answers and correct answers than less educated or informed people.

– those who agree to be in a study may be in some way different from those who refuse to participate (Volunteers may be different from those who are enlisted)

Information (Measurement) Bias

– a systematic difference between the measurements

(or information) recorded in different study groups.

– For example, in cohort studies, people with the risk

factor may be tested more frequently and carefully

than the control group. This is also called

‘surveillance’ bias or ‘diagnostic suspicion’ bias.

Interviewer Bias• an interviewer’s knowledge may influence the

structure of questions and the manner of presentation, which may influence responses

• Interviewer’s IQ may also influence in understanding the responses

• Observer Bias – observers may have preconceived expectations of what they should find in an examination

Recall Bias– a type of information bias, when people with a

certain condition are more likely to remember exposure to the risk factor under study than the control group. It can occur easily in case-control or cross-sectional studies, but not in cohort studies. (those with a particular outcome or exposure may remember events more clearly or amplify their recollections)

– For example parents of children with cancer may ‘remember’ more information about details of risk factors and their exposure to them, than parents of control children with identical exposure rates.

Attrition Bias (Loss to follow-up)

– Attrition is a reduction in the number of patients who remain in the study (patient drop out).

– This results in an attrition bias when the patients who drop-out of the study are systematically different from those who remain in and complete the study.

– It can occur in clinical trials and in cohort studies.

Types of Variables

Dependent Variables Independent Variable

Confounding Variable

Clearly state which variable is the dependent and which are the independent ones.

From the statement of the problem and the objectives of the study, determine whether a variable is dependent or independent.

When designing a study

Certain variables may produce changes in the DV which are mistakenly interpreted as representing effects of the IV. When this happens, the variable is called a confounding variable because its effects are confused with IV effects.

DefinitionA variable that is associated with the problem and with a possible cause of the problem is a potential Confounding Variable.

A confounding variable may either strengthen or weaken the apparent relationship between the problem and a

possible cause.

Cause Effect/outcome(independent variable) (dependent variable)

Other factors (confounding variables)

47

Confounding can be thought of as mixing of the

effect of the exposure under study on the

disease with that of an extraneous factor.

This external factor or variable must be

associated with the exposure and, independent

of the exposure must be a risk factor for the

disease.

Confounding

48

Example of confounding

Smoking MI

age

49

Table 1. Relation of Myocardial infarction (MI)to Recent Oral Contraceptive (OC) Use

MI Control Estimated relative risk

OC

Yes 29 135 =1.68

No 205 1607

Total 234 1742

50

Table: Age -specific Relation of Myocardial infarction (MI) to recent Oral Contraceptive (OC) Use

Age (yrs) Recent OC use

MI Controls Estimated age-Specific relative risk

25 – 29 Yes

No

4

2

62

2247.2

30 – 34 Yes

No

9

12

33

3908.9

35 – 39 Yes

No

4

33

26

330

1.5

40 – 44 Yes

No

6

65

9

362

3.7

45 – 49 Yes

No

6

93

5

301

3.9

Total 234 1742

51

Confounding can be controlled in study design through:

Restriction Matching exposure

Randomization

Confounding can be controlled in analysis through: Stratification Multivariate analysis

Estimation

The process of using sample information to draw conclusion about the value of a population parameter is known as estimation.

• A point estimate is a specific numerical value estimate of a parameter.

• The best point estimate of the population mean µ is the sample mean

• But how good is a point estimate?• There is no way of knowing how close the point

estimate is to the population mean• Statisticians prefer another type of estimate

called an interval estimate

Point Estimate

X

54

Interval Estimate• An interval estimate of a parameter is an interval or a

range of values used to estimate the parameter

Confidence Level The confidence level of an interval estimate of a parameter

is the probability that the interval estimate will contain the parameter

Three commonly used confidence levels are 90%, 95% and 99%

If one desires to be more confident then the sample size must be larger

What is Hypothesis?• A testable theory, or statement of belief used in

evaluation of a population parameter of interest e.g. Mean or proportion.

• Prediction of the relationship between one or more factors and the problem under study, which can be tested.

• Suppose a study is being conducted to answer

questions about difference between two regimens for the management of diarrhea in children:– The sugar based modern ORS and – The time-tested indigenous herbal solution made from

locally available herbs.

• One question that could be asked is:

"In the population is there a difference in overall improvement (after three days of treatment) between the ORS and the herbal solution?"

• There could be only two answers to this question:

• Yes • No

Null Hypothesis

"There is no difference between the 2 regimens in term of improvement” (null hypothesis).

A null hypothesis is usually a statement that there is no difference between groups or that one factor is not dependent on another and corresponds to the No answer.

Alternative Hypothesis

• "There is a difference in terms of improvement achieved by a three days treatment with the ORS and that of the herbal solution" (alternative hypothesis).

• Associated with the null hypothesis there is always another hypothesis or implied statement concerning the true relationship among the variables or conditions under study if no is an implausible answer. This statement is called the alternative hypothesis and corresponds to the “Yes” answer.

Types of Alternate Hypothesis

o Directional

o Non Directional

Court Decision Guilty Guilty Wrong Decision

Correct Decision Alpha Error (Type 1 error)

Wrong Decision Correct Decision

Not Guilty Beta Error (Type 2 error)

True SituationNot Guilty

αα and and ββ error error

When we mistakenly reject the null when indeed the null is true, then the type of wrong decision is known as a Type I or Alpha error.

However when we mistakenly do not reject our null when in fact it is false then we commit another error known as the Type II or Beta error.

P stands for probability. • Probability of rejecting Ho Hypothesis

when it is true.

P – Value

P – Value Also defined as • Probability of falsely rejecting Ho• Probability of finding the result by chance

alone• Probability of committing Type I error or

Alpha error

The P value is a function of two factors:•The magnitude of difference between

the groups.

•The size of the sample.

Steps in Hypothesis Testing

1. Statement of research question in terms of statistical hypothesis (Null and alternate hypothesis)

2. Selection of an appropriate level of significance. The significance level is the risk we are willing to take that a sample which showed a difference was misleading. (5% significance level means that we are ready to take a 5% chance of wrong results).

3. Choosing an appropriate statistics t test, z test for continuous data, chi square for proportions etc.

Test statistics is computed from the sample data and is used to determine whether the null hypothesis should be rejected or retained.

Test statistics generates p value

Steps in Hypothesis Testing

P value: Indicates the probability or likelihood of obtaining a result at least as extreme as that observed in a study by chance alone, assuming that there is truly no association between exposure and outcome under consideration.

By convention the p value is set at 0.05 level. Thus any value of p less than or equal to 0.05 indicates that there is at most a 5% probability of observing an association as large or larger than that found in the study due to chance alone given that there is no association between exposure and outcome. If p value0.05 do not reject the null hypothesis .

4. Performing calculations and obtaining p value.

5. Drawing conclusions, rejecting null

hypothesis if the p value is less than the set significance level.

SAMPLE SIZE ESTIMATIONSAMPLE SIZE ESTIMATION

Requirements

1 – the parameters (proportion or mean)

2 – the degree of precision (α)

3 – the desired confidence level (z)

4 – the estimated degree of variability of observation

(standard deviation)

Example 1Estimating The Sample Size

Question: What Parameter do you want to study?Answer: I want to estimate the average increase in body weights of Infant Rats, given Treatment A within a certain period of time.

Q 2 What is expected Std Dev

• S.D. = 20 gm• Q 3: How much error is tolerable?• ANSWER A : 10 gm OR less• ANSWER B : 5% of mean• Q 4: What is mean?• ANSWER: Mean = 100 gm• How confident do you want to be ?• 95% = {Z}

IN CASE OF ANS a TO Q3Sample Size will be

n = (Z)2 (S)2

e 2

= 1.96 x 1.96 x 20 x 20 = 16 10 x 10

IN CASE OF ANS b TO Q3 Sample Size will be

n = (Z)2 (S)2

e 2

= 1.96 x 1.96 x 20 x 20 = 62 5X5

76

Sample size estimationSample size estimation:

n = Z 2 Pq e

Z = 1.96 deviant error for .05

P = Prevalence

q = 1 - P

e = Tolerance error (5)(.01 ---> .1)

In Case of Prevalence as 20%Sample Size will be

n = Z2 x p x q e2

n = 1.96 x 1.96 x 20 x 80 = 246 5 x5

1. Type of study.

2. Magnitude of the outcome of interest derived

from previous studies.

3. Type of statistical analysis

required (comparing means or proportions).

4. Level of significance / Power.

Sample size calculations depend on:

1. The prevalence of the condition/ attribute of interest.

2. Level of confidence.

3. Margin of error.

Sample size for single proportion depends on:

Example of Sample size calculation for single proportion

A local health department wishes to estimate the prevalence of tuberculosis among children under 5 year of age in a locality. How many children should be included in the sample so that the prevalence may be estimated within 5% point of the true value with 95% confidence, if it is known that the true rate is unlikely to exceed 20%?

Sample size calculation and formula for single proportion

1. The Mean of the condition of interest.

2. Level of confidence.

3. Margin of error.

Sample size for single group mean depends on:

Example of Sample size calculation for single group mean

• A district medical officer seeks to estimate the mean hemoglobin level among pregnant women in his district. A previous study of pregnant women showed average hemoglobin level 8.2 g/dl and standard deviation of 4.2 g/dl. Assuming a sample of pregnant women is to be selected, how many pregnant women must be studied if he wanted the estimate should fall within 1 g/dl with 95% confidence?

Sample size calculation and formula for single group mean

Where ε =d/μ

1. The prevalence of the condition / attribute of interest for both groups.

2. Level of confidence.

3. Power of the test.

Sample size for two proportions depends on:

Example of Sample size calculation for two proportions

• It is believed that the proportion of patients who develop complications after undergoing one type of surgery is 5% while the proportion of the patients who develop complications after a second type of surgery is 15%. How large should the sample size be in each of the two groups of patients if an investigator wishes to detect with a power of 90%, whether the second procedure has a complication rate significantly higher than the first at the 5% level of significance?

Sample size calculation and formula for two proportions

1. The means/variance for both groups.

2. Level of confidence.

3. Power of the test.

Sample size for two group means depends on:

Example of Sample size calculation for two group means

Suppose the true mean systolic blood pressure (SBP) of 35 to 39 year old OC users is (132.86 mmHg) and standard deviation (15.34 mmHg). Similarly, for non-OC users, the mean SBP is (127.44 mmHg) with standard deviation (18.23 mmHg). If we desire to estimate the difference between 2 groups of equal size, what would be the minimal sample size required with a power of 80% at 95% confidence level?

90

Calculator

Sample size - Calculation

1. The prevalence of the condition/attribute of interest.

2. Estimated sensitivity.

3. Estimated specificity.

4. Level of significance.

5. Margin of error.

Sample size for sensitivity and specificity depends on:

Example of Sample size calculation for sensitivity and specificity

If we want to determine the sensitivity and specificity of graded compression ultrasonography in the diagnosis of acute appendicitis by the gold standard histopathology. How many patients should be included in the sample .The prevalence OF AA is 77% and estimated sensitivity of US is 96.5% and estimated specificity is 94.1% with 95% confidence, if we want to keep margin of error as 10%?

94

Sample size calculation and formula for sensitivity and specificity studies

Suggested websites for sample size calculators

1.http://www.raosoft.com/samplesize.html

2.http://www.quantitativeskills.com/sisa/calculations/samsize.htm

3.http://www.openepi.com/Menu/OpenEpiMenu.htm