basic statistics. content data types descriptive statistics graphical summaries distributions ...

Basic Statistics

Content

Data Types

Descriptive Statistics

Graphical Summaries

Distributions

Sampling and Estimation

Confidence Intervals

Hypothesis Testing (Statistical tests)

Errors in Hypothesis Testing

Sample Size

Data Types

Motivation

Defining your data type is always a sensible first consideration

You then know what you can ‘do’ with it

Variables

Quantitative Variable A variable that is counted or measured on a

numerical scale

Can be continuous or discrete (always a whole number).

Qualitative Variable A non-numerical variable that can be classified into

categories, but can’t be measured on a numerical scale.

Can be nominal or ordinal

Continuous Data

Continuous data is measured on a scale.

The data can have almost any numeric value and can be recorded at many different points.

For example:

Temperature (39.25oC)

Time (2.468 seconds)

Height (1.25m)

Weight (66.34kg)

Discrete Data

Discrete data is based on counts, for example:

The number of cars parked in a car park

The number of patients seen by a dentist each day.

Only a finite number of values are possible e.g. a dentist could see 10, 11, 12 people but not 12.3 people

Nominal Data

A Nominal scale is the most basic level of measurement. The variable is divided into categories and objects are ‘measured’ by assigning them to a category.

For example,

Colours of objects (red, yellow, blue, green)

Types of transport (plane, car, boat)

There is no order of magnitude to the categories i.e. blue is no more or less of a colour than red.

Ordinal Data

Ordinal data is categorical data, where the categories can be placed in a logical order of ascendance e.g.;

1 – 5 scoring scale, where 1 = poor and 5 = excellent

Strength of a curry (mild, medium, hot)

There is some measure of magnitude, a score of ‘5 – excellent’ is better than a score of ‘4 – good’.

But this says nothing about the degree of difference between the categories i.e. we cannot assume a customer who thinks a service is excellent is twice as happy as one who thinks the same service is good.

Motivation

Why important?

– extremely useful for summarising data in a meaningful way

– ‘gain a feel’ for what constitutes a representative value and how the observations are scattered around that value

– statistical measures such as the mean and standard deviation are used in statistical hypothesis testing

Session Content

Measures of Location

Measures of Dispersion


Measures of location

• Mean

• Median

• Mode

The average is a general term for a measure of location; it describes a typical measurement

Mean

The mean (arithmetic mean) is commonly called the average

In formulas the mean is usually represented by read as ‘x-bar’

The formula for calculating the mean from ‘n’ individual data-points is;

n

xx

x-bar equals the sum of the data divided by the number of data-points

Median

Median means middle

The median is the middle of a set of data that has been put into rank order

Specifically, it is the value that divides a set of data into two halves, with one half of the observations being larger than the median value, and one half smaller

18 24 29 30 32

Half the data > 29Half the data < 29

Mode

The mode represents the most commonly occurring value within a dataset

Rarely used as a summary statistic

Find the mode by creating a frequency distribution and tallying how often each value occurs

If we find that every value occurs only once, the distribution has no mode.

If we find that two or more values are tied as the most common, the distribution has more than one mode


Range

Interquartile range

Variance

Standard deviation

Measures of spread

The spread/dispersion in a set of data is the variation among the set of data values

They measure whether values are close together, or more scattered

Length of stay in hospital (days)Length of stay in hospital (days)4 162 6 8 10 12 14 42 6 8 10 12

Range

Difference between the largest and smallest value in a data set

The actual max and min values may be stated rather than the difference

The range of a list is 0 if all the data-points in the list are equal

4 16 DaysRange

Interquartile range

Measures of spread not influenced by outliers can be obtained by excluding the extreme values in the data set and determining the range of the remaining values

Interquartile range = Upper quartile – Lower quartile

4 Days

Interquartile Range

209 12Q1 Q3

Variance

Spread can be measured by determining the extent to which each observation deviates from the arithmetic mean

The larger the deviations, the larger the variability

Cannot use the mean of the deviations otherwise the positive differences cancel out the negative differences

Overcome the problem by squaring each deviation and finding the mean of the squared deviations = Variance

Units are the square of the units of the original observations e.g. kg2

Standard Deviation

The square root of the variance

It can be regarded as a form of average of the deviations of the observations from the mean

Stated in the same units as the raw data

Standard Deviation (SD)

Smaller SD = values clustered closer to the mean

Larger SD = values are more scattered

Days8 1210

1 SD1 SD Mean

4 16

Mean

10

1 SD1 SD

6 8 12 14

Variance & Standard Deviation

The following formulae define these measures

Population Sample

22

22

22

1

ss

n

xxs

N

x

Deviation Standard Deviation Standard

VarianceVariance

Variation within-subjects

If repeated measures of a variable are taken on an individual then some variation will be observed

Within-subject variation may occur because:

– the individual does not always respond in the same way (e.g. blood pressure)

– of measurement error

E.g. readings of systolic blood pressure on a man may range between 135-145 mm Hg when repeated 10 times

Usually less variation than between-subjects

Variation between-subjects

Variation obtained when a single measurement is taken on every individual in a group

Between-subject variation

E.g. single measurements of systolic blood pressure on 10 men may range between 125-175 mm Hg

Much greater variation than the 10 readings on one man

Usually more variation than within-subject variation

Session Summary



Graphical Summaries

Motivation

Why important?

– extremely useful for providing simple summary pictures, ‘getting a feel’ for the data and presenting results to others

– used to identify outliers

Session Content

Bar Chart

Pie Chart

Box Plot

Histogram

Scatter Plot

Displaying frequency distributions

Qualitative or Discrete numerical data can be displayed visually in a:

– Bar Chart

– Pie Chart

Continuous numerical data can be displayed visually in a:

– Box Plot

– Histogram

Bar Chart

Horizontal or vertical bar drawn for each category

Length proportional to frequency

Bars are separated by small gaps to indicate that the data is qualitative or discrete

Example: Bar Chart

Pie Chart

A circular ‘pie’ that is split into sections

Each section represents a category

The area of each section is proportional to the frequency in the category

Example: Pie Chart

What could improve this chart?

Box Plot

Sometimes called a ‘Box and Whisker Plot’

A vertical or horizontal rectangle

Ends of the rectangle correspond to the upper and lower quartiles of the data values

A line drawn in the rectangle corresponds to the median value

Whiskers indicate minimum and maximum values but sometimes relate to percentiles (e.g. the 5th and 95th percentile)

Outliers are often marked with an asterix

Example: Box Plot

Histogram

Similar to a bar chart, but no gaps between the bars (the data is continuous)

The width of each bar relates to a range of values for the variable

Area of the bar proportional to the frequency in that range

Usually between 5-20 groups are chosen

Example: Histogram

‘Shape’ of the frequency distribution

The choice of the most appropriate statistical method is often dependent on the shape of the distribution

Shape can be:

– Unimodal – single peak

– Bimodal – Two peaks

– Uniform – no peaks, each value equally likely

Unimodal data

When the distribution is unimodal it’s important to assess where the majority of the data values lie

Is the data:

– Symmetrical (centred around some mid-point)

– Skewed to the right (positively skewed) – long tail to the right

– Skewed to the left (negatively skewed) – long tail to the left

Displaying two variables

If one variable is categorical, separate diagrams showing the distribution of the second variable can be drawn for each of the categories

Clustered or segmented bar charts are also an option

If variables are numerical or ordinal then a scatter plot can be used to display the relationship between the two

Example: Scatter Plot

2520151050

80

70

60

50

40

30

20

10

0

Time on Diet

Weig

ht

Loss

Scatterplot of Weight Loss vs Time on Diet

Fitting the Line

If the scatter plot of y versus x looks approximately linear, how do we decide where to put the line of best fit?

By eye?

A standard procedure for placing the line of best fit is necessary, otherwise the line fitted to the data would change depending on who was examining the data

Regression

The least-squares regression method is used to achieve this

This method minimises the sum of the squared vertical differences between the observed y values and the line i.e. the least-squares regression line minimises the error between the predicted values of y and the actual y values

The total prediction error is less for the least-squares regression line than for any other possible prediction line

Example: Scatter Plot with Regression Line

2520151050

80

70

60

50

40

30

20

10

0

Time on Diet

Weig

ht

Loss

Scatterplot of Weight Loss vs Time on Diet

Weight Loss = 1.69 + 3.47 Time on Diet

Session Summary

Bar Chart

Pie Chart

Box Plot

Histogram

Scatter Plot

Distributions

Motivation

Why important?

– if the empirical data approximates to a particular probability distribution, theoretical knowledge can be used to answer questions about the data

– Note: Empirical distribution is the observed distribution (observed data) of a variable

– the properties of distributions provide the underlying theory in some statistical tests (parametric tests)

– the Normal Distribution is extremely important

Important point

It is not necessary to completely understand the theory behind probability distributions!

It is important to know when and how to use the distributions

Concentrate on familiarity with the basic ideas, terminology and perhaps how to use statistical tables (although statistical software packages have made the latter point less essential)

Normal Distribution

Used as the underlying assumption in many statistical tests

Bell-shaped

Symmetrical about the mean

Flattened as the variance increases (fixed mean)

Peaked as the variance decreases (fixed mean)

Shifted to the right if mean increases

Shifted to the left if mean decreases

Mean and Median of a Normal Distribution are equal

≈ 3 standard deviations (3 )

Intervals of the Normal Distribution

99.7%

95%

68%

Other distributions

t-distribution

distribution

F- distribution

2χ

Motivation

Why important?

– studying the entire population in the majority of cases is impractical, time consuming and/or resource intensive

– samples are used in studies to estimate characteristics and draw conclusions about the population

Populations and Samples

Population – the entire group of individuals in whom we are interested

E.g.

– All season ticket holders at Newcastle United

– All students at the University of Newcastle upon Tyne

– The entire population of the UK

– All patients with a certain medical condition

Sample – any subset of a population

Sampling

Samples should be ‘representative’ of the population Some degree of sampling error will exist when the

whole population is not used Asking people to choose a ‘representative’ sample is

subjective as people will choose differently. An objective method for selecting the samples is

desirable – a sampling strategy The advantage of sampling strategies are that they

avoid subjectiveness and bias

Sampling Strategies

Include:

Simple Random Sampling (SRS) Systematic Sampling Cluster Sampling Stratified Random Sampling

Simple Random Sampling

Sample chosen so that every member of a

population has the same chance (probability) of being included in the sample

To carryout Simple Random Sampling a list of all the sample units in the population is required (a sampling frame)

Each unit is assigned a number and ‘n’ units are selected from the population

Simple Random Sampling

Advantage SRS is a fairly simple and effective method of

obtaining a random sample from a population

Disadvantages It can theoretically result in an unbalanced sample

that does not truly represent some sector of the population.

It can be an expensive way to sample from a population which is spread out over a large geographic area

Point Estimates

It is often required to estimate the value of a parameter of a population e.g. the mean

Can estimate the value of the population parameter using the data collected in the sample

The estimate is referred to as the point estimate of the parameter as opposed to an interval estimate which takes a range of values

Sampling variation

If repeated samples were taken from a population it is unlikely that the estimates of the population (e.g. estimates of the mean) would be identical in each sample

However, the estimates should all be close to the true value of the population and similar to one other

By quantifying the variability of these estimates, information can be obtained on the precision of the estimate and sampling error can be assessed

In medical studies, usually only one sample is taken from a population, as opposed to many

Have to make use of the knowledge of the theoretical distribution of sample estimates to draw inferences about the population parameter

Sampling distribution of the mean

Many repeated samples of size n from a population can be drawn

If the mean of each sample was calculated a histogram of the means could be drawn; this would show the sampling distribution of the mean

It can be shown that:

– the mean estimates follow a Normal Distribution whatever the distribution of the original data (Central Limit Theorem)

– if the sample size is small, the estimates of the mean follow a Normal Distribution provided the data in the population follow a Normal Distribution

– the mean of the estimates equals the true population mean

Sampling distribution of the mean

– The variability of the distribution is measured by the standard error of the mean (SEM)

– The standard error of the mean is given by:

– where is the population standard deviation and n is the sample size

n

σ SEM

σ

Best estimates in reality

When we have only one sample (as is the usual reality), the best estimate of the population mean is the sample mean and the standard error of the mean is given by:

where s is the standard deviation of the observations in the samples and n is the sample size

n

s SEM

Interpreting standard errors

A large standard error means that the estimate of the population mean is imprecise

A small standard error means that the estimate of the population mean is precise

A more precise estimate of the population mean can be obtained if:

– the size of the sample is increased

– the data is less variable

Using SD or SEM

SD, the standard deviation, is used to describe the variation in the data values

SEM, the standard error of the mean, is used to describe the precision of the sample mean

– should be used if you are interested in the mean of data values

Motivation

Why important?

– used to provide a measure of precision for a population parameter such as the mean

– can be used in statistical tests as a method of testing whether the results are clinically important


The standard error is not by itself particularly useful

It is more useful to incorporate the measure of precision into an interval estimate for the population parameter – this is known as a confidence interval

The confidence interval extends either side of the point estimate by some multiple of the standard error

A 95% Confidence Interval

A 95% confidence interval for the population mean is given by:

If the study were to be repeated many times, this interval would contain the true population mean on 95% of occasions

Usual interpretation: the range of values within which we are 95% confident that the true population lies – although not strictly correct

n

s.xμ

n

s.x

961961

Interpretation of CI intervals

A wide interval indicates that the estimate for the population parameter is imprecise, a narrow one indicates that the estimate is precise

The upper and lower limits provide a means of assessing whether the results of a test are clinically important

Can check whether a hypothesised value for the population parameter falls within the confidence interval

Hypothesis Testing

Motivation

Why important?

– used to quantify a belief against a particular hypothesis (a statistical test is performed)

e.g. the hypothesis is that the rates of cardiovascular disease are the same in men and women in the population

– a statistical test could be conducted to determine the likelihood that this is correct, making a decision based on statistical evidence as to whether the hypothesis should be rejected or not rejected

Hypothesis Testing

Once data is collected a process called Hypothesis Testing is used to analyse it

There are specific types of hypothesis tests

Five general stages for hypothesis testing can be defined:

Stages of Hypothesis Testing

1. Define the Null & Alternative Hypotheses under study

2. Collect data

3. Calculate the value of the test statistic

4. Compare the value of the test statistic to values from a known probability distribution

5. Interpret the P-value and results

The Null Hypothesis

The Null Hypothesis is tested which assumes no effect (e.g. the difference in means equals zero) in the population

E.g. Comparing the rates of cardiovascular disease in men and woman in the population

Null Hypothesis H0: rates of cardiovascular disease are the same in men and woman in the population

The Alternative Hypothesis

The Alternative Hypothesis is then defined, this holds if the Null Hypothesis is not true

E.g. Alternative Hypothesis H1: rates of cardiovascular disease are different in men and woman in the population

Two-tail testing

In the previous example no direction for the difference in rates was specified

i.e. it was not stated whether men have higher or lower rates than woman

A two-tailed test is often recommended because the direction is rarely certain in advance, if one does exist

There are circumstance in which a one-tailed test is relevant

The test statistic

After data collection, the sample values are substituted into a formula, specific to the type of hypothesis test

A test statistic is calculated

The test statistic is effectively the amount of evidence in the data against H0

The larger the value (irrelevant of sign), the greater the evidence

Test statistics follow known theoretical probability distributions

The P-value

The test statistic is compared to values from a known probability distribution to obtain the P-value

The P-value is the area in both tails (occasionally one) of the probability distribution

The P-value is the probability of obtaining our results, or something more extreme, if the Null Hypothesis is true

The Null Hypothesis relates to the population rather than the sample

Use of the P-value

A decision must be made as to how much evidence is required to reject H0 in favour of H1

The smaller the P-value, the greater the evidence against H0

Conventional use of the P-value – rejecting H0

Conventionally, if the P-value < 0.05, there is sufficient evidence to reject H0

There is only a small chance of the results occurring if H0 is true

– H0 is rejected, the results are significant at the 5% level

Conventional use of the P-value – not rejecting H0

If the P-value > 0.05, there is insufficient evidence to reject H0

– H0 is not rejected, the results are not significant at the 5% level

NB: This does not mean that the null hypothesis is true, simply that we do not have enough evidence to reject it!

Using 5%

The choice of 5% is arbitrary, on 5% of occasions H0 will be incorrectly rejected when it is true (Type I error)

In some clinical situations stronger evidence may be required before rejecting H0

– e.g. rejecting H0 if the P-value is less than 1% or 0.1%

The chosen cut-off for the P-value is called the significance level of the test; it must be chosen before the data is collected

Parametric vs. Non-Parametric Tests

Hypothesis Tests which are based on knowledge of the probability distribution that the data follow are known as parametric tests

Often data does not conform to the assumptions that underly these methods

In these cases non-parametric tests are used

Non-Parametric Tests make no assumption about the probability distribution and generally replace the data with their ranks

Non-parametric tests

Useful when:

• sample size is small

• data is measured on a categorical scale (though can be used on numerical data as well)

However:

• they have less power of detecting a real difference than the equivalent parametric tests if all the assumptions underlying the parametric test are true

• they lead to decisions rather than generating a true understanding of the data

Statistical tests

Quantitative data, Parametric tests

– One-sample t-test

– Two-sample t-test

– Paired t-test

– One-way ANOVA

Statistical tests

Quantitative data, Non-parametric tests

– Sign test

– Wilcoxon signed ranks test

– Mann-Whitney U test

– Kruskal-Wallis test

Statistical tests

Qualitative data, Non-parametric tests

– z-test for a proportion

– McNemar’s test

– Chi-squared test

– Fisher’s exact test

Choosing a statistical test

Useful medical statistical books will contain a flowchart to help decide on the correct statistical test

Considerations include:

– Is the data quantitative or qualitative?

– How many groups of data are there?

– Can a probability distribution be assumed?

Examples

Paired t-test

Two sample t-test (paired)

Two samples related to each other and one numerical or ordinal variable of interest

E.g. in a cross-over trial, each patient has two measurements on the variable, one while taking treatment, one while taking a placebo

E.g. the individuals in each sample may be different but linked to each other in some way

Assumptions

The individual differences are Normally distributed with a given variance

A reasonable sample size has been taken so that the assumption of Normality can be checked

Assumptions not satisfied

If the differences do not follow a Normal distribution, the assumption underlying the t-test is not satisfied

Options:

– Transform the data

– Use a non-parametric test such as the Sign Test or Wilcoxon signed ranks test

Example

A peak expiratory flow rate (PEFR) was taken from a random sample of 9 asthmatics before and after a walk on a cold day

The mean of the differences before and after the walk = 56.11

The standard deviation of the differences = 34.17

Does the walk significantly influence the PEFR?

Example: Stages of a paired t-test

1) Define the Null and Alternative hypotheses under study:

Ho: the mean difference = 0

H1: the mean difference ≠ 0

Example: Stages of a paired t-test

2) Collect data before and after the walk

3) Calculate the value of the test statistic, t

4) Compare the value of the t statistic to values from the known probability distribution

5) The p-value = 0.001

A 95% confidence interval for the true difference is (29.8,82.4)

xt

926.4

917.34

011.56t

Paired t-test results

– there is strong evidence to reject the Null Hypothesis in favour of the Alternative Hypothesis

– there is strong evidence that the walk significantly effects PEFR, the difference ≠ 0

Paired Samples Statistics

323.8889 9 59.82567 19.94189

267.7778 9 50.00694 16.66898

Before Walk

After walk

Pair1

Mean N Std. DeviationStd. Error

Mean

Paired Samples Test

56.11111 34.17398 11.39133 29.84266 82.37956 4.926 8 .001Before Walk - After walkPair 1Mean Std. Deviation

Std. ErrorMean Lower Upper

95% ConfidenceInterval of the

Difference

Paired Differences

t df Sig. (2-tailed)

Mann-Whitney test

Mann-Whitney U test

The Mann-Whitney U test – two independent samples test

It is equivalent to the Kruskal-Wallis test for two groups

Mann-Whitney tests that two sampled populations are equivalent in location

Methodology

The observations from both groups are combined and ranked, with the average rank assigned in the case of ties

If the populations are identical in location, the ranks should be randomly mixed between the two samples

The test calculates the number of times that a score from group 1 precedes a score from group 2 and the number of times that a score from group 2 precedes a score from group 1

Example

Two samples of diastolic blood pressure were taken

Is there a difference in the population locations without assuming a parametric model for the distributions?

The equality of population means is tested through the use of a Mann-Whitney test

Are the two populations significantly different?

Example - Mann-Whitney U test

Ranks

8 7.50 60.00

9 10.33 93.00

17

Group1.00

2.00

Total

Diastolic BloodPressure 1

N Mean Rank Sum of Ranks

Test Statisticsb

24.000

60.000

-1.156

.248

.277a

Mann-Whitney U

Wilcoxon W

Z

Asymp. Sig. (2-tailed)

Exact Sig. [2*(1-tailedSig.)]

DiastolicBlood

Pressure 1

Not corrected for ties.a.

Grouping Variable: Groupb.

- there is no evidence to reject the Null Hypothesis in favour of the Alternative Hypothesis, p-value = 0.277 >0.05

- there is no evidence of a difference in blood pressure medians

Motivation

Why important?

– when interpreting the results of a statistical test, there is always a probability of making an erroneous conclusion (however minimal)

– it is important to ensure that these probabilities are minimised

– possible mistakes are called Type I and Type II errors

Type I error

Rejecting the Null Hypothesis when it is true

Concluding that there is an effect when in reality there is none

The maximum chance of making a Type I error is denoted by alpha α

α is the significance level of the test, we reject the null hypothesis if the p-value is less than the significance level

Type II error

Not rejecting the Null Hypothesis when it is false

Concluding that there is no effect when one really exists

The chance of making a Type II error is denoted by beta β

Its compliment 1- β, is the power of the test

Power of the test

The Power is the probability of rejecting the Null Hypothesis when it is false

i.e. the probability of making a correct decision

The ideal power of the test is 100%

However there is always a possibility of making a Type II error

Sample Size

Motivation

Why important?

– if the sample size is too small, there may be inadequate test power to detect an important existing effect/difference and resources will be wasted

– if the sample size is too large, the study may be unnecessarily time consuming, expensive and unethical

– have to determine a sample size which strikes a balance between making a Type I or Type II error

– an optimal sample size can be difficult to establish as an estimate of the results expected in the study is required

Calculating an optimal sample size for a test

The following quantities need to be specified at the design stage of the investigation in order to calculate an optimal sample size:

– The Power

– Significance level

– Variability

– Smallest effect of interest

Summary

Data Types


Graphical Summaries

Distributions



Hypothesis Testing (Statistical tests)


Sample Size

Book Reference

Medical Statistics at a Glance, 3rd Edition

(Aviva Petrie & Caroline Sabin)

ISBN: 978-1-4051-8051-1

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