Lecture 1 Interpretation of data
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Transcript of Lecture 1 Interpretation of data
RDP Statistical Methods in Scientific Research - Lecture 1 1
Lecture 1
Interpretation of data
1.1 A study in anorexia nervosa
1.2 Testing the difference between the samples
1.3 Confidence intervals for treatment effects
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1.1 A study in anorexia nervosa
Ben-Tovim, Whitehead and Crisp (1979)
“Sufferers from anorexia nervosa, even those whose bodieshave become severely emaciated, often maintain that their bodily dimensions are quite normal”
Are anorexics able to judge their own bodily dimensions?
Are they worse at doing so than healthy controls?
8 anorexics and 11 controls participated in a study of thesequestions
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The apparatus
Two lights on a horizontal beam
Move them together: “Say stop when the distance apart is the same as the width ofyour waist”
Repeat while they move apart, and then average the twomeasurements to give the perceived width
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Body perception index
Let
A= mean BPI for anorexics
C= mean BPI for controls
Null hypothesis is H0: A= C
perceived widthBPI 100
actual width
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The data
Anorexics: 130, 194, 160, 120, 152, 144, 120, 141
Controls: 202, 140, 168, 160, 147, 133, 229, 172, 130, 206, 153
Summary:
Overall (n = 19):mean = 157.95, standard deviation = 30.689
Anorexics (nA = 8):mean = 145.13, standard deviation = 24.398
Controls (nC = 11):mean = 167.27, standard deviation = 32.426
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Formulae
mean:
standard deviation (a measure of the spread of the data):
1 nx ... xx
n
2 2
1 nx x ... x xS
n 1
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Notes
Here we have means for anorexics and for controls and standard deviations SA for anorexics and SC for controls
These are sample means and sample standard deviations: they vary from sample to sample
The population means are A for anorexics and C for controls and the population standard deviations are A for anorexics and C for controls: these are fixed truths that will never be known precisely
and are estimates of A and C SA and SC are estimates of A and C
Ax Cx
Ax Cx
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1.2 Testing the difference between the samples
The two group means are different from one another
Are they significantly different?
Or might the difference just be due to chance?
We will use a t-test to find out
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The t-statistic
where
A C
A C
x xt
1 1S
n n
2 2A A C C
A C
n 1 S n 1 SS
n n 2
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Notes
We begin with an estimate of the difference between the means:
This is standardised by dividing by S: S2 is a weighted average of
Standardisation ensures that t is unit-free
Division by is a matter of convention,
but it does ensure that values are not too greatly affected by sample sizes
A C
1 1
n n
A Cx x
2 2A CS and S
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Calculation
so that
145.13 167.27t 1.622
1 129.377
8 11
2 27 24.398 10 32.426S 29.377
17
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Theory
Suppose that
the BPIs of anorexics follow the normal distribution with mean A and standard deviation
the BPIs of controls follow the normal distribution with mean C and the same standard deviation
Then, if A = C, the statistic t follows Student’s t-distributionon 17 degrees of freedom (17 = 19 – 2 = n # parameters)
a similar shape (slightly fatter) centred on 0
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The t-distribution
The probability that a random variable following Student’s t-distribution on 17 degrees of freedom is 1.627 is 0.061
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Interpretation
If the null hypothesis H0: A= C is true (and the populationshave the same standard deviation), then t is unusually negative
The chance of it being so negative (or even more so) is 0.061
This is the p-value against the one-sided alternative H1: A< C
The value 0.061 is not so small that one would wish to reject H0
and conclude that there is a significant difference – it shows atrend, but does not constitute strong evidence
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Caution!
The investigators sought evidence that anorexics had a poorer perception of their bodily dimensions than controls
– that A> C
The trend is in the opposite direction!
“So maybe the anorexics have a better perception, being so obsessed by their bodies”
Investigators are going to wish to interpret the data, whicheverdirection the difference, so use a two-sided p-value
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Two-sided p-value
Double the one-sided p-value to give the two-sided p-value:p = 0.122
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Convention
A two-sided p-value 0.05 is usually taken to represent strong evidence of an effect
This goes back to Fisher in the 1930s
It is rather arbitrary, but it is a useful yardstick
A one-sided p-value 0.025 is usually taken to represent strong evidence of an effect – this avoids “cheating” by choosing the direction of the difference once the data have been observed
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1.3 Confidence intervals for treatment effects
We have used A to denote the population mean of the BPIsfor anorexics and A to denote their population standarddeviations
These are estimated by the sample mean = 145.13 and bythe sample standard deviation SA = 24.398 respectively
How good an estimate of A is 145.13?
How big or small might A actually be?
A confidence interval will answer this question
Ax
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Another t-distributed random variable
Let
Note that you cannot calculate tA as it depends on the unknown A
If the BPI observations are normally distributed, then tA followsStudent’s t-distribution with (nA – 1) df
Now, a t7 random variable lies between 2.365 and 2.635 withprobability 0.95
A A AA
A
x nt
S
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A confidence interval for A
It follows that, with probability 0.95,
which is
which is
which is
A A A
A
x n2.365 2.365
S
A A A A A A2.365 S n x 2.365 S n
A A A A A A Ax 2.365 S n x 2.365 S n
A145.13 2.365 24.398 8 145.13 2.365 24.398 8
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A confidence interval for A
So, with probability 0.95,
We say that (124.73, 165.53) is a 95% confidence interval for A
The upper and lower limits are random, while A is fixed
The limits capture the true value of A with probability 0.95
It could well be that the true mean BPI for anorexics is as low as124.73, it could also be as high as 165.53
A124.73 165.53
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A confidence interval for C
For the controls, nC = 11, = 167.27 and SC = 32.426
The 97.5% point of the t distribution on 10 df is 2.228
Hence, the 95% confidence interval for C is
(167.27 2.228 32.426/11) (145.49, 189.05)
Note that the confidence intervals for A and C overlap
What about a 95% confidence interval for = A C?
Cx
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A confidence interval for = A C
Now
follows Student’s t-distribution with (n – 2) df
The 97.5% point of the t distribution on 17 df is 2.110
A C
A C
x xt
1 1S
n n
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A confidence interval for = A C
It follows that, with probability 0.95
so that the 95% confidence interval for is
A C A CA C A C
1 1 1 1x x 2.110 S x x 2.110 S
n n n n
A CA C
1 1x x 2.110 S
n n
1 122.14 2.110 29.377
8 11
50.94,6.66
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Interpretation
0 lies within the confidence interval consistent with lack of significant evidence against
H0: = 0 at the 5% level (2-sided), as found from the t-test
The mean BPI for anorexics could be substantially lower that thatfor controls (by more than 50), or slightly higher
Larger sample sizes would reduce the width of the confidenceintervals, and make it easier to determine whether there really is adifference