Statistics for Neurosurgeons

A David MendelowBarbara A Gregson

Newcastle upon TyneEngland, UK

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The Normal Distribution

Means and standard errors

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

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

Displaying data

• Bar chart• Pie chart

• Histogram

• Box and whisker• Scatterplot

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

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

Boxplot (Box and Whisker Plot)

Plot of volume of haematoma by age group in STICH).

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).

Summarising data

• Central tendency– Mean– Median– Mode

• Spread– Range– Interquartile range– Standard deviation/variance

Confidence intervals

– statistic ± (1.96 x standard error)

– e.g. difference between means ± (1.96 x standard error of difference)

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

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

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

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

Comparison of related values: a.Linear regression (best

linear fit)

Linear regression (best linear fit)

Comparison of related values: b.Altman Bland Plots

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

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

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

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

Multivariate techniques

• Multiple linear regression• Logistic regression• Survival analysis

– Kaplan Meier– Cox proportional hazard model

Survival Functions

days

2101801501209060300

Pro

bab

ility

of s

urvi

val

1.0

.9

.8

.7

.6

Early Surgery

Initial Conservative Treatment

KaplanMeierPlot ofSurvival

Type I and type II errors

Null hypothesis

False True

Test result

Significant

Power(1-)

Type I error ()

Not significant

Type II error ()

ROC Curves

• Multiple chi squared 2 x 2 tests• See www.

Figure 1: ROC curve for % change in SJVO2 as a predictor of clinical ischaemia during awake carotid endarterectomy

Multiple 2x2 tables = ROC Curve