Download Non Normal Distribution - Faculty of Health, Education and Life

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Background to Statistics
for
non-statisticians
Craig Jackson
Prof. Occupational Health Psychology
Faculty of Education, Law & Social Sciences
BCU
[email protected]
Keep it simple
“Some people hate the very name of statistics but.....their power of
dealing with complicated phenomena is extraordinary. They are the
only tools by which an opening can be cut through the formidable
thicket of difficulties that bars the path of those who pursue the science
of man.”
Sir Francis Galton, 1889
How Many Make a Sample?
How Many Make a Sample?
“8 out of 10 owners who expressed a preference, said their cats
preferred it.”
How confident can we be about such statistics?
8 out of 10?
80 out of 100?
800 out of 1000?
80,000 out of 100,000?
Types of Data / Variables
Continuous
Discrete
BP
Height
Weight
Age
Children
Age last birthday
colds in last year
Ordinal
Nominal
Grade of condition
Positions 1st 2nd 3rd
“Better- Same-Worse”
Height groups
Age groups
Sex
Hair colour
Blood group
Eye colour
Conversion & Re-classification
Easier to summarise Ordinal / Nominal data
Cut-off Points
(who decides this?)
Allows Continuous variables to be changed into Nominal variables
BP
> 90mmHg
=
Hypertensive
BP
=< 90mmHg
=
Normotensive
Easier clinical decisions
Categorisation reduces quality of data
Statistical tests may be more “sensational”
Good for summaries
Bad for “accuracy”
BMI
Obese vs Underweight
Types of statistics / analyses
DESCRIPTIVE STATISTICS
Describing a phenomena
Frequencies
Basic measurements
How many…
Meters, seconds, cm3, IQ
INFERENTIAL STATISTICS
Inferences about phenomena
Hypothesis Testing
Confidence Intervals
Correlation
Significance testing
Proving or disproving theories
If sample relates to the larger population
Associations between phenomena
e.g diet and health
Multiple Measurement
or….
why statisticians and love don’t mix
25 cells
22 cells
26
25
24
24 cells
23
22
21
21 cells
20
Total
Mean
SD
= 92 cells
= 23 cells
= 1.8 cells
Small samples spoil research
N
Age
IQ
N
Age
IQ
N
Age
IQ
1
2
3
4
5
6
7
8
9
10
20
20
20
20
20
20
20
20
20
20
100
100
100
100
100
100
100
100
100
100
1
2
3
4
5
6
7
8
9
10
18
20
22
24
26
21
19
25
20
21
100
110
119
101
105
113
120
119
114
101
1
2
3
4
5
6
7
8
9
10
18
20
22
24
26
21
19
25
20
45
100
110
119
101
105
113
120
119
114
156
Total
Mean
SD
200
20
0
1000
100
0
Total
Mean
SD
216
21.6
± 4.2
1102
110.2
± 19.2
Total
Mean
SD
240
24
± 8.5
1157
115.7
± 30.2
Central Tendency
Mode
Median Mean
Patient comfort rating
10
9
8
31
27
Frequency
70
7
6
5
4
3
2
1
121
140
129
128
90
80
62
Dispersion
Range Spread of data
Mean
Arithmetic average
Median Location
Mode
Frequency
SD
Spread of data
about the mean
Range 50-112 mmHg
Mean 82mmHg
SD
± 10mmHg
Median 82mmHg
Mode
82mmHg
Dispersion
An individual score therefore possess a standard deviation (away from the
mean), which can be positive or negative
Depending on which side of the mean the score is
If add the positive and negative deviations together, it equals zero
(the positives and negatives cancel out)
central value (mean)
negative deviation
positive deviation
Dispersion
Range
The interval between the highest and lowest measures
Limited value as it involves the two most extreme (likely faulty) measures
Percentile
The value below / above which a particular percentage of values fall
(median is the 50th percentile)
e.g 5th percentile - 5% of values fall below it, 95% of values fall above it.
A series of percentiles (1st, 5th, 25th, 50th, 75th, 95, 99th) gives a good general
idea of the scatter and shape of the data
1st 5th
25th
50th
75th
95th 99th
Range
5’6”
5’7”
5’8”
5’9”
5’10”
5’11”
6’
6’1”
6’2”
6’3”
6’4”
Standard Deviation
To get around this, we square each of the observations
Makes all the values positive (a minus times a minus….)
Then sum all those squared observations to calculate the mean
This gives the variance - where every observation is squared
Need to take the square root of the variance, to get the standard deviation
SD
=
 Σ x2 – (Σ x)2 / N
(N – 1)
Grouped Data
Normal Distribution
SD is useful because of the shape of many distributions of data.
Symmetrical, bell-shaped / normal / Gaussian distribution
Non Normal Distribution
Some distributions fail to be symmetrical
If the tail on the left is longer than the right,
the distribution is negatively skewed (to the left)
If the tail on the right is longer than the left, the
distribution is positively skewed
(to the right)
Normal Distributions
Standard Normal Distribution has a mean of 0 and a standard deviation of 1
The total area under the curve amounts to 100% / unity of the observations
central value (mean)
3 SD
2 SD
1 SD
0 SD
1 SD
2 SD
3 SD
Proportions of observations within any given range can be obtained from the
distribution by using statistical tables of the standard normal distribution
95% of measurements / observations lie within 1.96 SD’s either side of the
mean
Quincunx machine 1877
balls dropped through a
succession of metal pins…..
…..a normal distribution
of balls
do not have a normal
distribution here. Why?
Normal & Non-normal distributions
The distribution derived from the
quincunx is not perfect
It was only made from 18 balls
Distributions
Sir Francis Galton (1822-1911) Alumni of Birmingham University
9 books and > 200 papers
Fingerprints, correlation of calculus, twins, neuropsychology, blood
transfusions, travel in undeveloped countries, criminality and meteorology)
% of population
Deeply concerned with improving standards of measurement
5’6”
5’7”
5’8”
5’9”
5’10” 5’11”
Height
6’
6’1”
6’2” 6’3”
6’4”
Normal & Non-normal distributions
Galton’s quincunx machine ran with hundreds of balls
a more “perfect” shaped normal distribution.
Obvious implications for the size of samples of populations used
The more lead shot runs through the quincunx machine, the smoother the
distribution
in the long run . . . . .
Presentation of data
Table of means
Exposed
n=197
Controls
n=178
Age
(yrs)
45.5
( 9.4)
I.Q
105
( 10.8)
Speed 115.1
(ms) ( 13.4)
T
P
48.9
( 7.3)
2.19
0.07
99
( 8.7)
1.78
0.12
94.7
( 12.4)
3.76
0.04
Presentation of data
Category tables
Exposed
Controls
Healthy
50
150
200
Unwell
147
28
175
197
178
375
Chi square (test of association) shows:
Chi square = 7.2
P = 0.02
Bar Charts
A set of measurements can be presented either as a table or as a figure
Graphs are not always as accurate as tables, but portray trends more easily
Title of graph
y-axis
Legend key
y-axis
label
(ordinate)
Data display area
scale
x-axis (abscissa)
groups
Bar Charts
Some Real Data
A combination of distributions is acceptable to facilitate comparisons
Movie goers’ ratings for both movies
7000
Vacation
6000
Empire
Votes
5000
4000
3000
2000
1000
0
1
2
3
4
5
6
7
User rating
8
9
10
With a scatter diagram, each
individual observation becomes a
point on the scatter plot, based on two
co-ordinates, measured on the
abscissa and the ordinate
ordinate
Correlation and Association
abscissa
Two perpendicular lines are drawn through the medians - dividing the plot into
quadrants
Each quadrant should outlie 25% of all observations
Correlation and Association
Correlation is a numerical expression between 1 and -1 (extending through all points
in between). Properly called the Correlation Coefficient.
A decimal measure of association (not necessarily causation) between variables
Correlation
of
1
Maximal - any value
of
one
variable
precisely determines
the other. Perfect +ve
Correlation of -1 Any value of one
variable precisely determines the other,
but in an opposite direction to a
correlation of 1. As one value increases,
the other decreases. Perfect -ve
Correlation of 0 - No
relationship between
the variables. Totally
independent of each
other. “Nothing”
Correlation of 0.5 - Only a slight
relationship between the variables i.e
half of the variables can be predicted
by the other, the other half can’t.
Medium +ve
Correlations between 0 and 0.3 are weak
Correlations between 0.4 and 0.7 are moderate
Correlations between 0.8 and 1 are strong
Correlation and Association
Correlation is a numerical expression between 1 and -1 (extending through all points
in between).
Properly called the Correlation Coefficient.
A decimal measure of association (not necessarily causation) between variables
How can the above variables be correlated?
Sampling Keywords
POPULATIONS
Can be mundane or extraordinary
SAMPLE
Must be representative
INTERNALY VALIDITY OF SAMPLE
Sometimes validity is more important than generalizability
SELECTION PROCEDURES
Random
Opportunistic
Conscriptive
Quota
Sampling Keywords
THEORETICAL
Developing, exploring, and testing ideas
EMPIRICAL
Based on observations and measurements of reality
NOMOTHETIC
Rules pertaining to the general case (nomos - Greek)
PROBABILISTIC
Based on probabilities
CAUSAL
How causes (treatments) effect the outcomes
Clinical Research
Types of clinical research
Experimental vs. Observational
Longitudinal vs. Cross-sectional
Prospective vs. Retrospective
Experimental
Longitudinal
Observational
Longitudinal
Cross-sectional
Prospective
Prospective
Retrospective
Randomised Controlled Trial
Cohort studies
Case control studies
Survey
Experimental Designs
Between subjects studies
Treatment group
Outcome measured
Control group
Outcome measured
patients
Within Subjects studies
patients
Outcome measured #1
Treatment
Outcome measured #2
Observational studies
Cohort (prospective)
cohort
prospectively measure risk factors
end point measured
aetiology
prevalence
development
odds ratios
Case-Control (retrospective)
start point measured
aetiology
odds ratios
prevalence
development
retrospectively measure risk factors
cases
Case-Control Study – Smoking & Cancer
“Cases” have Lung Cancer
“Controls” could be other hospital patients (other disease) or “normals”
Matched Cases & Controls for age & gender
Option of 2 Controls per Case
Smoking years of Lung Cancer cases and controls
(matched for age and sex)
Cases
n=456
Smoking years 13.75
(± 1.5)
Controls
n=456
6.12
(± 2.1)
F
7.5
P
0.04
Cohort Study: Methods
Volunteers in 2 groups e.g. exposed vs non-exposed
All complete health survey every 12 months
End point at 5 years: groups compared for Health Status
Comparison of general health between users and non-users of mobile
phones
ill
healthy
mobile phone user
292
108
400
non-phone user
89
313
402
381
421
802
Randomized Controlled Trials in GP & Primary Care
90% consultations take place in GP surgery
50 years old
Potential problems
2 Key areas:
Recruitment Bias
Randomisation Bias
Over-focus on failings of RCTs
RCT Deficiencies
Trials too small
Trials too short
Poor quality
Poorly presented
Address wrong question
Methodological inadequacies
Inadequate measures of quality of life (changing)
Cost-data poorly presented
Ethical neglect
Patients given limited understanding
Poor trial management
Politics
Marketeering
Why still the dominant model?
Quantitative Data Summary
• What data is needed to answer the larger-scale research question
• Combination of quantitative and qualitative ?
• Cleaning, re-scoring, re-scaling, or re-formatting
• Measurement of both IV’s and DV’s is complex but can be simplified
• Binary measurement makes analysis easier but less meaningful
• Binary data needs clear parameters e.g exposed vs controls
• Collecting good quality data at source is vital
Quantitative Data Summary
•
Continuous & Discrete data can also be converted into Binary data
•
Normal distribution of participants / data points desirable
•
Means - age, height, weight, BMI, IQ, attitudes
•
Frequencies / Classifications - job type, sick vs. healthy, dead vs alive
•
Means must be followed by Standard Deviation (SD or ±)
•
Presentation of data must enhance understanding or be redundant
If you or anyone you know has been
affected by any of the issues
covered in this lecture, you may
need a statistician’s help:
www.statistics.gov.uk
Further Reading
Abbott, P., & Sapsford, R.J. (1988). Research methods for nurses and the
caring professions. Buckingham: Open University Press.
Altman, D.G. (1991). Designing Research. In D.G. Altman (ed.), Practical
Statistics For Medical Research (pp. 74-106). London: Chapman and Hall.
Bland, M. (1995). The design of experiments. In M. Bland (ed.), An introduction
to medical statistics (pp5-25). Oxford: Oxford Medical Publications.
Bowling, A. (1994). Measuring Health. Milton Keynes: Open University Press.
Daly, L.E., & Bourke, G.J. (2000). Epidemiological and clinical research
methods. In L.E. Daly & G.J. Bourke (eds.), Interpretation and uses of medical
statistics (pp. 143-201). Oxford: Blackwell Science Ltd.
Jackson, C.A. (2002). Research Design. In F. Gao-Smith & J. Smith (eds.), Key
Topics in Clinical Research. (pp. 31-39). Oxford: BIOS scientific Publications.
Further Reading
Jackson, C.A. (2002). Planning Health and Safety Research Projects. Health
and Safety at Work Special Report 62, (pp 1-16).
Jackson, C.A. (2003). Analyzing Statistical Data in Occupational Health
Research. Management of Health Risks Special Report 81, (pp. 2-8).
Kumar, R. (1999). Research Methodology: a step by step guide for beginners.
London: Sage.
Polit, D., & Hungler, B. (2003). Nursing research: Principles and methods (7th
ed.). Philadelphia: Lippincott, Williams & Wilkins.