Download - Statistics for Neurosurgeons A David Mendelow Barbara A Gregson Newcastle upon Tyne England, UK
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Statistics for Neurosurgeons
A David MendelowBarbara A Gregson
Newcastle upon TyneEngland, UK
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The Normal Distribution
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Means and standard errors
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Comparison of curves(Sig. dif vs. Sig. bigger)
• Bell shaped: Student’s t test• Paired data: Student’s paired t
test• Skewed curves: Non parametric
tests– Sign test (+ve –ve)– Wilcoxson ranked sum test
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Types of data
• Binary eg Yes/No, Male/Female
• Nominal eg eye colour (blue/green/brown)
• Ordinal eg normal/weak/paralysed, GCS eye
• Counts no. of aneurysms, no. of operations
• Continuous width of haematoma
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Displaying data
• Bar chart• Pie chart
• Histogram
• Box and whisker• Scatterplot
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Bar Chart and Pie Chart
Total GCS at randomisation in STICH II Figures for the first 234 cases
Median GCS=13
Males, 54%
Females, 46%
Gender of patients in STICH II Figures for the first 234 cases
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Histograms
Figures produced on 19/11/2009: 234 cases
Mean = 63.8 Std = 12.85Median = 65 yearsQuartiles = 55, 74Min = 20 years, Max = 94 years
Mean = 39.5 Std = 21.44Median = 35mlQuartiles = 22, 54Min =10ml, Max =96ml
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Boxplot (Box and Whisker Plot)
Plot of volume of haematoma by age group in STICH).
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Scatterplot
Plot of 1,490 simultaneous end tidal and arterial CO2 measurements. Dot areas are proportional to
the number of measurements with that combination of values. End tidal CO2 values tend to be
lower than corresponding PaCO2 values (most points are below the equivalence line).
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Summarising data
• Central tendency– Mean– Median– Mode
• Spread– Range– Interquartile range– Standard deviation/variance
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Confidence intervals
– statistic ± (1.96 x standard error)
– e.g. difference between means ± (1.96 x standard error of difference)
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Comparison of means
• Sample mean v population mean– One sample t-test
• Two small sample means– T-test (assuming equal variance)– T-test (assuming unequal variance)
• Two paired samples means– Paired t-test
• Large samples– Z-test
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Comparison of tables (2x2)
• Fisher’s exact testp = (r1!r2!c1!c2!)/n!a!b!c!d!
• Chi Squared testObserved vs. expected frequencies
a b r1
c d r2
c1 c2 n
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Chi squared testa b r1
c d r2
c1 c2 n
• McNemar’s = (a - d)2/(a + d) • degrees-of-freedom = (rows - 1)
(columns - 1) = 1
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Relative risk sensitivity and specificity
Test +ve Test -ve
Disease yes a b r1
Disease no c d r2
• Sensitivity = a/r1• Specificity = d/r2
• Positive predictive value = a/a+c• Negative predictive value = d/b+d
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Comparison of related values: a.Linear regression (best
linear fit)
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Linear regression (best linear fit)
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Comparison of related values: b.Altman Bland Plots
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Statistical tests comparing two samples
• Binary – Large frequencies – χ2, compare proportions, odds ratio– Small frequencies – Fisher’s exact
• Nominal not ordered– Large frequencies – χ2, – Small frequencies – combine categories
• Nominal ordered– Large frequencies – χ2 for trend
• Ordinal– Mann-Whitney U test
• Continuous – Large samples – Normal distribution for means– Small normal samples – Two sample t test– Small non normal – Mann-Whitney U test
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Statistical tests for paired or matched data
• Binary McNemar
• Nominal Stuart test
• Ordinal Sign test
• Continuous (small, non-normal) Wilcoxonmatched pairs
• Continuous (small, normal) Paired t-test
• Continuous (large) Normal distribution
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Choice of test for independent observations
Outcome variable
Nominal Categ >2 Catrg Ordered
Ordinal Non-normal
Normal
Input variable
Nominal χ2
Fisher
χ2 χ2 trend
Mann-Whitney
Mann-Whitney
Mann-WhitneyLog rank
Student’s tNormal test
Categ >2 χ2 χ2 χ2 Kruskal-Wallis
Kruskal-Wallis
Analysis of variance
Categ Ordered
χ2 trend
Mann-Whitney
χ2 Kendall’s rank
Kendall’s rank
Kendall’s rank
Kendall’s rank
Linear regression
Ordinal Logistic regression
Kruskal-Wallis
Kendall’s rank
Spearman rank
Spearman rank
Spearman rank Linear regression
Non-normal
Logistic regression
Kruskal-Wallis
Kendall’s rank
Spearman rank
Spearman rank
Spearman rank and
linear regression
Normal Logistic regression
Logistic regression
Spearman rank
Spearman rank
Spearman rank and
linear regression
Pearson and Linear
regression
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Relative risk and odds ratios
With disease Without disease
Male a b r1
Female c d r2
• Risk for men p1 = a/r1• Risk for women p2 = c/r2
– Relative risk = p1/p2• Odds for men = a/b• Odds for women = c/d
– Odds ratio = (a/b)/(c/d) = ad/bc
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Multivariate techniques
• Multiple linear regression• Logistic regression• Survival analysis
– Kaplan Meier– Cox proportional hazard model
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Survival Functions
days
2101801501209060300
Pro
bab
ility
of s
urvi
val
1.0
.9
.8
.7
.6
Early Surgery
Initial Conservative Treatment
KaplanMeierPlot ofSurvival
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Type I and type II errors
Null hypothesis
False True
Test result
Significant
Power(1-)
Type I error ()
Not significant
Type II error ()
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ROC Curves
• Multiple chi squared 2 x 2 tests• See www.
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Figure 1: ROC curve for % change in SJVO2 as a predictor of clinical ischaemia during awake carotid endarterectomy
Multiple 2x2 tables = ROC Curve