Reminder of 1st lecture

1
Reminder of 1st lecture

• Data types
• Summarising qualitative data counts,
• Summarising quantitative data
• Location mean, median
• Spread standard deviation, interquartile
  range
• Graphical presentation of data

2
Qualitative data
Dichotomous (Binary) This variable has only 2
possible categories (mutually exclusive) Nominal
This variable has more than 2 categories,
mutually exclusive and unordered Ordinal This
variable has more than 2 categories, mutually
exclusive and ordered
3
Quantitative data
Continuous This is used for something measured
on a scale. The variable can take any value
within a range of values e.g. Height in cm,
Weight in kg. Discrete This variable often
represents counts (integer values) e.g. Age to
nearest year, number of children
4
Measures of central tendency

• Mean - arithmetic average (? mu)
• add up all observations and divide by the number
  of observations
• Median - central value of the distribution
• rank observations and the median is the
  observation below which 50 of all values fall

5
Histogram

• Assessment of normality

6
Appropriate measures of location and spread
7
Reminder of 2nd lecture

• The probability for an event or outcome indicates
  how likely it is to happen.
• The probability of an event has to be between 0
  and 1.
• A probability of 0 means that the event never
  happens.
• A probability of 1 means that an event always
  happens.
• Events that cannot happen together are mutually
  exclusive and their probabilities can be added
  together. P(A or B) P(A) P(B)
• Events that are independent do not affect each
  other and their probabilities can be multiplied
  together. P(A and B) P(A) x P(B)

8
Properties of Normal curve
P0.68 P0.95 P0.999
9
Chi-Squared

• Another important distribution related to the
  normal is the Chi-squared distribution.
• Its use is used when investigating categorical
  data.

10
Crossstab Example

• If eye and hair colour are not associated then
  for example, the Expected number with blue eyes
  and blond hair would be

11

• So the chi-squared is found by looking a function
  of the discrepancy between observed and expected
  counts in each cell
• summed over all combinations of hair and eye
  colour.
• If this is large and in the tail of the
  distribution, then it may be said that the
  observed is not as expected!
• More of this later.

12
Summary

• Probabilities are integral to all things around
  us.
• We can derive and understand probabilities.
• We have seen that probabilities build together to
  form probability distributions.
• Some are theoretical distributions that are well
  understood, the most important being the Normal.
• Using these theoretical distributions we can
  begin to make inferences about the population on
  the basis of samples.

13
Sampling, sampling distributions and statistical
inference

• Gordon Prescott

14
Wednesday, 11 October 2006Scots bar staff health
'improved'

• The health of Scotland's bar staff has improved
  dramatically since the introduction of a smoking
  ban, a medical study has found.
• Researchers at Dundee University found
  significant health improvements in the first two
  months after the March ban.
• The results have led to calls for the UK
  Government to speed the introduction of a similar
  ban south of the border.
• But smokers' rights group Forest said the link
  between passive smoking and ill health had not
  been proven.
• The team from the university's asthma and allergy
  research group began testing bar workers in and
  around Dundee in February, a month before the ban
  came into force.
• Using a series of indicators, they established
  symptoms attributable to passive smoking,
  measuring lung function and inflammation in the
  bloodstream.
• This study provides compelling evidence that
  making workplaces smoke free can have a
  significant and speedy impact on people's
  health Peter Hollins, British Heart
  Foundation

15
Study of bar workers in Dundee before and after
the smoking ban

• JAMA. 20062961742-1748 To investigate the
  association of smoke-free legislation with
  symptoms, pulmonary function, and markers of
  inflammation of bar workers (n77)
• At 1 month The percentage of bar workers with
  respiratory and sensory symptoms decreased from
  79.2 (n  61) before the smoke-free policy to
  53.2 (n  41)
• (total change 26 95 confidence interval CI,
  13.8 to 38.1 Plt0.001)
• Forced expiratory volume in the first second
  increased from 96.6 predicted to 104.8
• (change 8.2 95 CI, 3.9 to 12.4 Plt0.001)
• Serum cotinine levels decreased from 5.15 to 3.22
  ng/mL
• (change 1.93 ng/mL 95 CI, 2.83 to 1.03
  ng/mL Plt0.001)

16
Study of bar workers in Dundee before and after
the smoking ban

• JAMA. 20062961742-1748
• At 2 months Total white blood cell reduced from
  7610 to 6980 cells/µL
• (630 cells/µL 95 CI, 1010 to 260 cells/µL
  P  0.002)
• Neutrophil count was reduced from 4440 to 4030
  cells/µL
• (410 cells/µL 95 CI, 740 to 90 cells/µL
  P  0.03)
• Smoke-free legislation was associated with
  significant early improvements in symptoms,
  spirometry measurements, and systemic
  inflammation of bar workers

17
Sampling theory

• A population is the totality of observations
  obtainable from all subjects possessing some
  common specified characteristic
• male diabetics
• height of all females in the UK
• A sample is a set of observations which
  constitutes part of a population

18
Random and biased samples

• Random sampling is a sampling technique where
  each member in the population is chosen entirely
  by chance, with a known chance of being included
  in the sample. Representative of the population.
• The most common random sample is obtained when
  any one individual or measurement in the
  population is as likely to be selected as any
  other.
• Can also have a biased sample where some
  individual or measurements have a greater chance
  of being included than others.
• With a non-random sampling technique not all
  members have a known chance of being selected, or
  some members have a zero chance of being
  selected. Unrepresentative of the population.

19
Selection of sample

• Probability sampling
• Each item/person in the population has a
  calculable non-zero chance of being selected for
  the sample.
• Sampling error (degree to which the sample
  differs from the population) can be calculated.
• Convenience sampling
• The items/people are selected by the researcher
  from the population in a non-random manner.
• The sampling error is unknown and cannot be
  calculated.

20

• Random sampling
• Simple Random sampling,
• Systematic Random sampling,
• Stratified Random sampling,
• Cluster Random sampling,
• Multi-Stage sampling.
• Biased sampling
• Quota sampling

21
Probability sampling

• Simple random sample
• Each item in the population has an equal chance
  of being selected for the sample

22
Random number table

• 84 42 56 53 87 75
• 78 87 77 03 57 09
• 85 86 48 86 12 39
• 65 37 93 76 46 11
• 09 49 41 73 76 49
• 64 06 71 99 37 06
• 46 69 31 24 33 52
• 67 85 07 75 56 96

23
Systematic random sampling

• Choose one element at random from the population
  of size N
• For a sample of size n, choose every N/n element
  thereafter

• Advantages - It is simpler and can be more
  representative than a simple random sample
• Disadvantages - possibility of implicit
  clustering, not a simple random sample

24
Example systematic random sampling

• A survey of GPs in Scotland to assess counselling
  services provided to patients is to be conducted.
• 1 in 3 GPs are to be included in the sample.
• List of GPs available is ordered by practice
  within Health Board.
• Systematic sample will ensure representative
  spread across health boards and practices.

25
Stratified random sampling

• The strata are subgroups of the population which
  are chosen to minimize differences between
  members of the same strata and maximize the
  differences between members of different strata.

• Main advantages
• Increases the representativeness of the sample
• Increases the precision of the resulting
  estimates
• Allows comparison between strata

26
Example Stratified random sampling

• It is likely that size of practice, site of
  practice (rural/urban) may influence whether the
  practice employs a counsellor
• The list of GPs could be stratified into number
  of partners (lt 4, gt 4) in the practice and area
  (rural/urban)
• A simple random sample from each of the four
  lists would constitute the sample
• The sample would ensure that each strata (or
  combination of strata) are represented in the
  sample

27
Cluster random sampling

• The clusters are subgroups of the population
  which are chosen to maximize differences between
  members of the same cluster and therefore
  minimize the differences between members of
  different clusters

• Advantages - Cheaper and faster than a simple
  random sample
• Disadvantages - Less representative than a simple
  random sample and there is a danger of
  contamination between respondents

28
Example Cluster random sampling

• For a nutritional survey to be carried out, 20
  schools in Scotland will be randomly selected.
• All secondary year 4 children from those schools
  selected will be interviewed.

29
Multi-stage sampling

• Different sampling units are sampled at different
  stages
• Example
• Geographical areas of the UK would randomly be
  selected, from which hospitals would be randomly
  selected from which wards/patients would then be
  randomly selected.

30
Convenience sampling

• Quota sampling
• In this type of sampling an individual is chosen
  by an interviewer.
• To avoid undue bias the quota is sub-divided into
  various categories e.g. male/female, old/young
  and so on.
• The interviewer is given quotas for each category
  and uses discretion to select the interviewees.

31
Statistical Inference
POPULATION
SAMPLE
INFERENCE
32

• Which sampling approach will lead to unbiased
  estimates?

33
Statistical Inference
POPULATION
SAMPLE
INFERENCE
34
Population parameters and sample statistics

• A population parameter is a measurable
  characteristic of the population
• Values obtained from a sample are estimates of
  the population parameters

35
Estimation I

• A parameter is a numerical descriptive measure of
  a population. It is calculated from the
  observations in the population
• mean - m
• standard deviation - s
• A sample statistic is a numerical descriptive
  measure of a sample. It is calculated from the
  observations in the sample
• Sample mean -
• Sample standard deviation - s

36
Estimation II

• Population parameters are fixed so long as the
  population itself does not change
• Sample statistics will vary from sample to
  sample, even though samples may be random and the
  population does not change

37
Sampling distribution

• In theory we could select all possible random
  samples from a population and gain an estimate of
  the population parameter from each of the
  individual samples.
• If a histogram of the sample estimates for each
  individual sample was plotted, this would form
  the sampling distribution of the population
  parameter (probability distribution).

38
Sampling Distribution
39
Sampling distribution

• The sampling distribution of a sample statistic,
  calculated from a sample of n measurements, is
  the probability distribution of the statistic

40
Example

• Imagine that a random sample of 100 individuals
  is to be selected from a population
• Their height in cm is measured
• The mean height is computed
• Another random sample of 100 individuals from the
  same population is taken
• Their height in cm in measured
• Their mean height is computed
• This is repeated until 20 random samples have
  been taken

41
20 samples of size 100

• The first sample of heights of 100 people gives a
  mean of 172.03 cm and a standard deviation (SD)
  of 6.03 cm.
• The second sample gives mean 173.50 cm SD 6.74
  cm.
• These figures represent the mean height (cm) for
  each of the 20 random samples
• 172.03 173.50 171.89 171.95 170.59
• 172.63 172.72 171.99 172.50 171.71
• 172.55 172.86 171.58 172.83 172.55
• 171.28 172.62 171.41 171.38 172.26

42
Histogram of means of 20 samples
43
Histogram of means of 100 samples
44
Sampling error

• Each random sample may have a different estimate
  of the population parameter due to sampling
  variation
• Knowledge of the sampling distribution allows us
  to assess how close the estimate obtained from
  one individual sample is to the true population
  parameter. This is known as precision.

45
Precision

• The larger the size of the sample the greater the
  reduction in sampling error
• Taking a larger sample will result in reducing
  the sampling variation from the true
  population value that we are trying to estimate.
• This implies that our estimate would be more
  precise.

46
Standard Error

• The standard deviation of the sampling
  distribution of the mean is known as the standard
  error of the mean.
• The standard error provides a measure of how far
  from the true population value the estimate is
  likely to be (the precision)

47
Standard deviation/standard error

• The standard deviation, s, is a measure of the
  variability of individuals in a sample
• The standard error is a measure of the
  uncertainty in the sample statistic (e.g. mean,
  proportion)

48
What does the standard error indicate?

• Consider again the random sample of 100
  individuals is to be selected from a population
• Their height in cm is measured
• The mean height is computed
• Another random sample of 100 individuals from the
  same population is taken
• Their height in cm in measured
• Their mean height is computed
• This is repeated until 20 random samples have
  been taken

49
20 samples of size 100

• These figures represent the mean height (cm) for
  each of the 20 random samples
• 172.03 173.50 171.89 171.95 170.59
• 172.63 172.72 171.99 172.50 171.71
• 172.55 172.86 171.58 172.83 172.55
• 171.28 172.62 171.41 171.38 172.26
• Mean of the 20 samples 172.14 cm
• SD of 20 sample means 0.689 ?SE (mean)

50
Explanation of sampling error

• 5 students in population
• Ages are 22, 25, 28, 30, 35
• Sample of 3 students randomly selected to
  estimate age in population (5 students)

51
22, 25, 28, 30, 35 ?28

• Sample 1 22,30,35 mean 29
• Sample 2 25,28,35 mean 29.3
• Sample 3 22,28,30 mean 27.7
• Sample 4 25,30,35 mean 30
• Sample 5 28,30,35 mean 31

NB Variation in age 22 to 35 Variation in mean
age 27.7 to 31
52
Relationship between sample size and precision of
sample estimate

• Heights
• Sample size mean SD Standard error
• 100 172.03 6.03 0.60
• 200 172.77 6.42 0.45
• 500 171.99 6.85 0.31
• 1000 172.15 6.84 0.22

53
Properties of the sampling distribution of the
mean

• The mean of the sampling distribution mean of
  the population distribution
• Standard deviation of the sampling distribution
  ? / ?n standard error of the mean
• The sampling distribution of the mean is
  approximately Normal for large sample sizes

54
Central Limit Theorem

• If a random sample of n observations is selected
  from a population, then when n is sufficiently
  large, the sampling distribution of (the
  mean) will be approximately a Normal
  distribution.
• The larger the sample size, the better the
  approximation to the Normal distribution will be.
• A sample size of at least 30 will usually be
  enough.

55
Statistical Inference
Representativeness Size
POPULATION
SAMPLE
INFERENCE
56
Statistical Inference

• Concerned with how we draw conclusions from
  sample data, about the larger population from
  which the sample is selected.
• There are two types of inference
• Confidence Intervals (Estimation)
• Hypothesis Testing (Significance Testing)

57
Confidence intervals

• When we collect data on a sample of individuals
  we would not expect the results from our sample
  to be exactly the same as those we would get if
  we had data on the whole population.
• Using the variability in the sample data we can
  calculate a range of values in which the
  population value is likely to lie.
• We can vary the width of this range depending on
  how confident we want to be that we will have
  included the true population value (usually set
  at 95 confidence).

58
Calculation of confidence intervals

• Confidence intervals can be calculated for most
  sample estimates using the following notation
• Sample estimate ? critical value x standard
  error(sample estimate)

59
Calculation of confidence interval for a
population mean

• Sample estimate
• sample mean
• Standard error of mean
• sample standard deviation/?n
• For large samples, 5 critical value 1.96
• Point estimate 172.03 cm SE 0.603 cm
• 95 CI (170.85 to 173.21) cm

60
Small samples

• For a small sample (nlt30) we use the
  t-distribution with (n-1) degrees of freedom
• 95 CI for small samples
• When n is large the t-distribution approximates
  to the Normal distribution
• For n 5, 10, 30, 60, 120,
• t n-1(5) 2.57, 2.23, 2.04, 2.00, 1.98

61
Interpretation of CIs

• Consider an infinite number of random samples of
  size n from a population
• A mean and 95 CI could be computed for each of
  the samples
• For 95 of the samples from the population, the
  95 CI will include the population value, whilst
  in 5 of samples the 95 CI will not include the
  population value.

62
Interpretation of confidence intervals

• Sample 1

Sample 2
Sample 3
Sample 4
Sample 5
Sample n
True population value (e.g. ?, ?)
63
Example

• Suppose I want to know the average height of UK
  people. I have a random sample of 100 people with
  a mean of 172.03 and SD of 6.03.
• The standard error of the mean is 6.03/?1000.603
• The estimate of the population mean is 172.03

• The 95 confidence interval (CI) for the
  population mean is 172.03 - 1.96 x 0.603, 172.03
  1.96 x 0.603
• 95 CI is 170.85, 173.21
• I am 95 confident that the average height of UK
  people is between 170.85 and 173.21

• A different 100 people would have a slightly
  different sample mean and confidence interval
• 95 of random samples of 100 would give a 95
  confidence interval containing the true
  population mean

64
100 samples of 100 people from a population of
10000 with known mean 172 cm and SD 6.8 cm
65
Change in sample size

• If the sample size is larger, the standard error
  is smaller and therefore the CI is also narrower
• For my first sample of 100 the standard error of
  the mean was 0.603 cm and 95 CI was 170.85,
  173.21 cm

• When I took a sample of 200 the mean was 172.77
  cm and the SD was 6.42 cm.
• The standard error was 6.42/?2000.45 (smaller)
• 95 CI was 172.77 - 1.96x0.45, 172.77 1.96x0.45
• 171.88, 173.65 cm

• For a sample of 1000 the 95 CI was 171.73,
  172.58 cm

66
Confidence intervals

• Raising the confidence level from 95 to 99
  increases the assurance that the confidence
  interval contains the population mean, but it
  makes the estimate less precise i.e. the width of
  the CI is wider
• Multiplier changes from 1.96 to 2.58.
• For height with 100 in the sample
• 95 CI (170.85, 173.21) cm
• 99 CI (170.47, 173.58) cm

67
Sample size

• It is possible to determine what sample size
  should be taken, if we wish to achieve a given
  level of precision
• This is because precision can be increased by
  reducing the size of the standard error
• The size of the standard error is based on the
  size of the sample

68
Practicals

• There are no new computer practicals this week.
• Instead there is the opportunity to complete the
  data description practical from last week and
  to ask questions in the Friday time slot.
• The worked example for the data description
  practical will be on the web early next week.