Day 2 Session 1

Basic Statistics

Cathy Mulhall

South East Public Health Observatory

Spring 2009

Overview

• Types of data

• Summarising data

• The Normal distribution

• Confidence intervals

• Hypothesis testing

• P-values

Types of Data

Numerical(Quantitative)

Counted or measured

Discrete Continuous

Categorical(Qualitative)

Characterises a quality

Nominal Ordered

Numerical data

Discrete

Integers (whole numbers)

Examples• Number of people• Number of teeth

Continuous

Any value on a scale

Examples• Height• Weight

Categorical data

Nominal

No natural order

Examples• Gender• Ethnic group

Ordered

Have a natural order

Examples• Socio-economic group• Cancer Staging (I – IV)

Which types of data are the following?

• Screening test result• Parity (no. of children) • Pain scale• Age at last birthday• Exact age• Alive at 6 months? • Number of bed days in

hospital

– categorical nominal– numerical discrete– categorical ordered– numerical discrete– numerical continuous– categorical nominal– numerical discrete

Summarising numerical data

1. Location (central tendency)

Mean, median, mode

2. Spread (variation)

Range, percentiles, standard deviation

Location

• Mean – sum of all obs / number of obs

• Median – value that divides the dist in 2,odd no. of obs - middle obs even no. of obs - mean of central pair

• Mode – value that occurs most frequently

Mean and median?

a) 3, 4, 5, 6, 7

b) 9,10, 20, 21

c) 1, 2, 3, 4, 990

Mean

25/5

60/4

1000/5

Median

5

15

3

Variation

• Range = highest value – lowest value

• Interquartile range = upper quartile – lower quartile (i.e. 3rd quartile – 1st quartile)

• Percentile – value below which a given proportion of the data lies

Variance

• Step 1: Calculate ‘Deviations’ = the difference between each observation and the mean of the data

• Step 2: Square these Deviations

• Step 3: Average the Squared Deviations

• … this is the Variance

• (Strictly, divide by n-1, not n)

Standard Deviation

• Step 4: Take the square root of the Variance (this returns the statistic to the same units as the data)

… this is the Standard Deviation

SD measures the amount of variability in the population

Summarising categorical data

• Percentages and rates

• Covered in Day 3 – Introduction to Analysis session

Normal distribution

• Symmetric

• Bell shaped

• Standard Normal Distribution

Mean = 0 SD = 1

• Represents the distribution of values observed if whole population was studied

Normal distribution

Mean, Median, Mode

Normal Distribution, changes in mean

Normal Distribution

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-3

-2.7

-2.5

-2.2

-1.9

-1.6

-1.4

-1.1

-0.8

-0.5

-0.3 0

0.2

7

0.5

5

0.8

2

1.1

1.3

7

1.6

5

1.9

2

2.2

2.4

7

2.7

5

3.0

2

3.3

3.5

7

3.8

5

4.1

2

4.4

Pro

bab

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y d

en

sit

y

μ=0.7μ=0 μ=2

σ=1

Normal Distribution, changes in SD

Normal Distribution

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-3

-2.8

-2.6

-2.3

-2.1

-1.9

-1.7

-1.4

-1.2 -1

-0.8

-0.5

-0.3

-0.1

0.1

2

0.3

5

0.5

7

0.8

1.0

2

1.2

5

1.4

7

1.7

1.9

2

2.1

5

2.3

7

2.6

2.8

2

Pro

bab

ilit

y d

en

sit

y

σ=1

σ=0.6

σ=0.3

μ=0

Normal distribution

• Defined by complex math formulae

• Published tables listing the area under the Standard Normal Curve

• Standard N scores – Z scores

• Used to calculate area between 2 points

• 95% of dist lies within +/- 2 SD of mean

• Known as ‘reference range’

Normal distribution

Importance of N distribution

• Many biological variables are N dist or can be made N dist by transformation

• Many statistical tests require data to be N distributed

• If data skewed need to transform

• 1/X, Log (X), sqrt (X)

Symmetric and Skewed Data

Population Samples

Bias

• Deviation from true result

• Minimised by random sampling

Random Error

• In any random sample there will be sampling variation

• Minimised by random sampling

Sampling Variability

Sampling Variability

Hypothesis Tests Confidence Intervals

Standard Error

• Standard deviation measures the amount of variability in the sample estimate

• It indicates how closely the population mean or proportion is likely to be to the sample estimate

Standard Error

n

SDxSE ,xMean,

n

pppSE

1pProportion,

Confidence Intervals

• Based on the Normal distribution, 95% sample estimates will be within 1.96 SEs from the true value

• For 95% of samples this interval will contain the true population value

• For any one sample there is a 95% chance that the interval contains the true value

xSEx 96.1 pSEp 96.1

Confidence Intervals

• 5% risk (or 1 in 20 chance) than true value lies outside the 95 % interval

• Tells us how imprecise our estimate is

• Provides a range of values within which the true (population) value is likely to lie

Narrow 95% CI precise estimate

Wide 95% CI imprecise estimate

0

5

10

15

20

25

30

35

Self-reported smoking status in women (%), by ethnic group with 95% confidence intervals (England, 2004)

• What can we say about the true smoking prevalence for the general population?

• For which ethnic groups is the prevalence of smoking significantly different from 25%?

• Is the prevalence of smoking significantly different between the Black Caribbean and Black African populations?

• Is the prevalence of smoking significantly different between the Pakistani and Bangladeshi populations?

Interpretation of confidence intervals

• 95% confident that the true smoking prevalence for the general population is between 22.5 and 24.5%

• For Black African, Indian, Pakistani, Bangladeshi and Chinese the prevalence of smoking is significantly different from 25%

• The prevalence of smoking is significantly different between Black Caribbean and Black African groups

• Cannot be sure that the prevalence of smoking is significantly different between the Pakistani and Bangladeshi populations

Interpretation of confidence intervals

Interpretation of confidence intervals

• Non overlapping intervals indicative of real differences

• Overlapping intervals need to be considered with caution

• Need to be careful about using confidence intervals as a means of testing.

• The smaller the sample size, the wider the confidence interval

Hypothesis Tests

• Assess strength of evidence for an association

• Test statistic calculated using population value, sample estimate and stnd. error

• Null hypothesis; no true difference between groups in population from which samples arose

Hypothesis Tests

• If the null hypothesis is true, what are the chances of getting as big (or bigger) as that observed

• Uses population value sample estimate and Standard Error

• Null hypothesis; no true difference between groups in population from which samples arose

Illustration of acceptance regions

Principles of Testing

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-3

-2.8

-2.6

-2.3

-2.1

-1.9

-1.7

-1.5

-1.2 -1

-0.8

-0.6

-0.3

-0.1 0.1

0.32

0.55

0.77 1

1.22

1.45

1.67 1.9

2.1

2.32

2.55

2.77 3

Pro

bab

ilit

y d

ensi

ty

accept null hypothesis

reject null hypothesis

reject null hypothesis

μ0

P-values

• probability of obtaining a difference as large (or larger) as that observed, if there is really no difference in the population from which the samples came, i.e. if the null hypothesis is true

P-values

Small p-value (p<0.05)

• unlikely that the sample arose for a pop where null is true

• Evidence for a real difference in pop

Large p-value (p>0.05)

• likely that the sample arose for a pop where null is true

• No evidence to reject the null hypothesis

Interpretation of P-values

Source; Essential medical statistics By Betty R. Kirkwood, Jonathan A. C. Sterne

Quiz

A person was defined as hypertensive if their diastolic blood pressure was > 90 mmHg & their systolic was > 140 mmHg. The variable ‘hypertensive’ is:

a) Paired continuousb) Nominal categoricalc) Skewedd) Continuous

What conclusion can be drawn from this figure?

a) The mean is less than the standard deviation

b) The mean is higher than the median

c) There are fewer observations below the mean than above it

d) The mean is approximately equal to the median

Based on a sample of 153 newborns, the 95% CI for the pop mean birth weight was between 3181 and 3319 grams:

a) 95% of the individual birth weights are between 3181 & 3319 grams

b) The true mean for the 153 newborns is probably between 3181 & 3319 grams

c) The mean of the population from which the 153 newborns came is between 3181 & 3319 grams

d) There is a 95% chance that the true mean of the population from which the 153 newborns came is included in the range 3181 - 3319 grams

Useful Resource

http://www.apho.org.uk/apho/techbrief.htm

Finding out more

www.healthknowledge.org.uk

Conclusions

• Cover some basic statistical concepts

• Gain insight into what they mean

• Gain confidence in understanding basic statistics

Basic Statistics Exercise• Exercise 1 - Calculate some summary

statistics for class size data in spreadsheet

• Exercise 2 – using the CI template provided calculate the 95% CI for the mean class size from exercise 1

Basic Statistics Exercise

• To download file go to http://www.sepho.org.uk and search on “intelligence training” then Day 2

or go tohttp://www.sepho.org.uk/viewResource.aspx?id

=12272

• Useful Excel FunctionsAVERAGE, MEDIAN, MODE, QUARTILE, PERCENTILE, VAR, STDEV