The Mathematics of Biostatistics

Chapter 6 and 7

Our Progress So Far

Week 1,2 and 3:• Examination of the

theory of epidemiology

• How this theory relates to biostatistics

Week 4:• Delving into the

numbers game of biostatistics

• How biostatistics related to epidemiology

Simplifying Statistics

To make a statistical operation more simple do the following:

• Write out the formula

• Plug in all the numbers in the appropriate places (make sure you have the right numbers)

• Work from the inside of the equation to the outside in terms of solving things

• Solve the equation, remember we are simply working with +, -, x, and ∙∕∙ all of your basic functions

Attributable Risk (AR)

Defined: An estimate of the amount of risk which is attributable to the risk factor

Formula:

AR = [a/(a+b)] – [c/(c+d)]

Problem # 1Refer to Pp. 92, Table 6-1

Using Table 6-1 (pp. 92) as our guide to what a,b,c, and d mean, lets use the data shown in Figure 6-1.

Therefore: • A = 191 (smokers dying of lung cancer)

• B = 999809 (100,000 population – A)

• C = 8.7 (non-smokers dying of lung cancer)

• D = 99991.30 (100,000 population – C)

Working the Equation

Step 1: • AR = [a/(a+b)] – [c/(c+d)]

Step 2:• AR = [191/(191+999809)] – [8.7/(8.7+99991.30)]

Step 3:

• 191-8.7

100,000

Step 4:• 182.3/100,000

ANY QUESTIONS ON THIS FORMULA?

Try One On Your Own

History: For a given year there was a heart attack death rate in people 50 pounds over their ideal weight of 1346 per 100,000 population. Among people of a normal weight there was a heart attack death rate in people within their ideal body weights of 200 per 100,000 population.• Please identify a,b,c and d

• Determine the AR (you have 3 minutes)

AR = [a/(a+b)] – [c/(c+d)]AR = [1346/(1346+98654)] – [200/(200+99800)]

AR = 1346 – 200/100,000AR = 1146/100,000

Relative Risk

Defined: This is somewhat of a comparison of the ratio of risk in an exposed group to the ratio of risk in the unexposed group.

• Formula:

RR = [a/(a+b)]/[c/(c+d)]

Hint: Notice that we are dividing the two sets of numbers not subtracting them as we did with AR

Use Data From Slide # 5

RR = [a/(a+b)] / [c/(c+d)]RR = [191/(191+999809)] / [8.7(8.7+99991.3)]

RR = [191/100,000] / [8.7/100,000]

RR = 191 / 8.7

100,000

RR = 21.95

100,000

RR = 22 (round up)

Your Turn

Using the information from our obesity example, solve for RR You have 3 minutes

Here are the values for your convenience• A = 1346

• B = 98654

• C = 200

• D = 99800

RR = [a/(a+b)] / [c/(c+d)]RR = [1346/(1346+98654)/[200/(200+99800)RR = 1346/200

100,000

RR = 6.73/100,000

? ? ? ? ?

So what does all of this data mean?• Slide 11 = Smokers are 22 times more likely

to die from lung cancer than non-smokers

• Slide 12 = People weighing 50 pounds over their ideal body weight are 7 times more likely to die from heart attacks than people within their normal weight range.

Ratio

Defined: An estimate of a odds ratio

Formula:OR = (a/c) / (b/d)

HINT: Do not use the step in the book that instructs you to convert the above formula to OR = ad/bc. The reason is because the numbers sometimes become too large to work with and muddy the waters.

Using Slide # 12 Data

OR = (a/c) / (b/d)

OR = (1346 / 200) / (98654 / 99800)

OR = 6.73 / .989

OR = 6.80

Your Turn…

Using the data from Slide # 5 solve for OR

Data is below for your convenience• A = 191

• B = 999809

• C = 8.7

• D = 99991.30

You have 3 minutes

OR = (a/c) / (b/d)OR = (191/8.7) / (999809/99991.30)OR= 21.95/10OR = 2.195 or 2.2

Attributable Risk Percent

Defined: A method of determining the total risk of death due to a condition found in the group practicing a particularly “risky” behavior.

FormulaAR%(exposed) = Riskex – Riskunex X 100

Riskex

Back to Slide 5 Data

AR% = Riskex – Riskunex x 100

Riskex

AR% = 191-8.7 x 100 191

AR% = 182.3 X 100191

AR% = 95.4

So…

According to this data, 95.4% of the lung cancer found in the smokers population is caused by the risk factor of smoking.

Key Concepts

Accuracy: Ability of a measurement to be correct on the average

Precision: Ability of a measurement to give the same results with repeated measurements of the same thing

Both of these are necessary in statistics and neither takes a back seat to the other

VariabilityWho looks can make all the difference…or none at all

Intraobserver variability = A difference of observation/interpretation of data when studied by the same person

Interobserver variability = A difference of observation/interpretation of data when studied by more than one person

False is False and True is True Or is it?

Type I Error• Also known as a false-positive error or Alpha

error

• The error is in the fact that a positive reading is registered when the results are actually negative

Continued…

Type II Error• Also known as a false-negative error or a beta

error

• The error is in the fact that a negative reading is registered when the results are actually positive

Sensitive Vs. Specific

Sensitivity – Ability of a test to detect the disease when present

Specificity – Ability of a test to indicate non-disease status when no disease is present

A Summary of Tonight’s Class

Mathematical manipulation of data Relationship between the data and the

population it was taken from Support of epidemiological reckoning

with statistical analysis of data

QUESTIONS

Future Plans

Utilize the statistical tools conquered tonight

Build on those tools with more tools

Become junior statisticians who can use statistics to understand epidemiological principles