Why statistics ?

1
Why statistics ?

• To understand studies in clinical journals.
• To design and analyze clinical research studies.
• Because of this, questions on statistics appear
  on board examinations.

2
Types of Clinical Research Studies

• Cohort all patients have some condition or
  something in common (e.g., healthy and living in
  Framingham, MA)
• Case-Control cases have some condition controls
  do not
• Randomized, placebo-controlled treatment trial
  all patients have the condition
• May be unblinded, single blinded or double
  blinded
• Randomized, active-treatment controlled trial
  all patients have the condition
• often phase 3 trial
• Meta analysis multiple studies of same
  condition, although definition of the condition
  may vary from study to study

3
Types of Variables

• CONTINUOUS
• AGE
• BP
• CRP
• AST, CK, glucose, etc
• HEIGHT
• WEIGHT
• BMI
• Etc.

• CATEGORICAL
• GENDER
• OBESE
• CURE
• MI
• RACE
• OLD vs YOUNG
• Etc.

4
Between subject variability Serum Na in 135
normals
Mean, 140 median 140 range, 135-145
mM standard deviation 2
5
Basic Statistical Terms

• Range the two extreme values (min and max)
• Mean the average value (uses all values)
• Median the middle value (ignores extreme
  values), which divides population into two
  subgroups
• Quartiles divides all values into 4 groups
• Tertiles, Quintiles, Percentiles
• Standard deviation measure degrees of difference
  among all values (uses all values)
• SD ?(?(differences from the mean2 )/n-1)

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Is there a volunteer ?
Values (n3) Difference from mean Differences2
12 ? ?
10 ? ?
8 ? ?
Mean? Median? ??

• /n-1 x/2?
• ?? ?
• SD ?

Mean SD ?
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The normal (bell-shaped) distribution

• Imagine 2 curves with the same mean, but
  different SDs ( one wider and less precise the
  other narrow-er and more precise).
• Now imagine two curves with different means and
  standard deviations from this curve
• Statistical tests are designed to tell us to what
  extent these different curves could have occurred
  by chance

mean
n
Standard deviations (SD) from the mean. 95 of
values are within 1.96 SD of mean
8
Some important statistical concepts

• Confidence intervals (usually reported as 95 CI)
• Number needed to treat (or harm)
• Absolute and relative risk or benefit reductions
  (or increases)
• 2-by-2 tables (Chi square, Fisher exact, Mantel
  Haenszel, others)
• Odds or hazard ratios
• Type 1 and 2 errors
• Estimating sample size needed for a study
• Pre- and post-test probabilities and likelihood
  ratios

Ann Int Med 2009 150 JC6-16
9
95 CI

• H. pylori eradication/NSAID study with outcome
  of ulcer or no ulcer (categorical outcome)
• 5 of 51 (10, or .10) Hp pts. who received
  antibiotics got ulcers when exposed to NSAID.
• and 15 of 49 (31, or .31) Hp pts. who did
  not receive antibiotics got ulcers when exposed
  to NSAID.
• What is the chance this difference in outcome
  occurred due to chance and not the antibiotics?

Lancet 2002 3599-13.
10
95 CIs

• The proportions, p1 and p2, of patients who
  got ulcers in the 2 groups are an estimate of the
  true rate. However, from this estimate we can be
  95 confident that the actual rates ranges from A
  to B, with p1 and p2 in the center of the
  interval from A to B. A and B are the 95
  confidence intervals.

p1
A
B
T H E 9 5 C O N F I D E N C E I N T E R V A L
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95 Confidence interval (CI)

• To calculate the 95 CI for p (i.e., A and B),
  use this formula
• p 1.96? (p)(1-p)/n

The larger the n, which is in the denominator,
the smaller (more precise) the CI
12

• 5 of 51 (p110, or .10) of the antibiotic group
  got ulcers when exposed to NSAID for a fixed time
• 95 CI .10 ? 1.96?(.1)(.9)/51.10.08.02,
  .18? 2,18
• 15 of 49 (p231, or .31) of the placebo- group
  got ulcers when exposed to NSAID for a fixed time
• 95CI .31?1.96?(.31)(.69)/49 .31.13.18,.44?
  18, 44

Note the two 95 CIs do not overlap, which means
that differences are unlikely to be due to
chance. But is the ARR significant?
13
Absolute risk reduction (ARR) and its 95 CI

• The ARR with antibiotics was 31 minus 10, or
  21.
• The 95 CI of the ARR
• 21 ? 1.96 ? (p1)(1-p1)/n1(p2)(1-p2)/n2)
  21 ?15, or 6, 36.
• The ARR with antibiotics is somewhere between 6
  and 36, with 95 confidence.
• This CI does not overlap zero and thus is
  unlikely due to chance.

14
Number needed to treat (NNT)

• If Absolute Risk reduction (ARR) 31-1021,
  the number needed to treat
  1/ARR 1/.215.
• Number needed to harm is the same concept as
  number needed to treat except that the
  intervention caused harm rather than good
• e.g. how many patients needed to be treated with
  antibiotics to produce one drug rash

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RRR

• Relative Risk Reduction (RRR) ARR/risk with
  placebo..
• In this example, RRR 21/31 68.
• Treat 1,000 pts. with NSAID? 310 ulcers (31)
• Treat 1,000 pts. with NSAID Abs? 100 ulcers
  (10)
• Antibiotic use prevented 210 ulcers (210/310
  68 RRR)
• Antibiotic use reduced ulcers from 310 to 100, or
  to 32
• of expected, a reduction of 68.
• Note Length of exposure to NSAID in this study
  in the 2 groups was identical. If two groups are
  not followed for an identical time, often the
  case in trials, outcomes may be higher in the
  group followed longer and thus events need to be
  expressed per unit of time (e.g., events per 100
  patient-years)

16
Another example, with the outcome of VTE or no
VTE (categorical outcome)

• 14 of 255 (p15.5, or .055) patients with VTE
  switched to low-intensity warfarin developed
  another VTE
• 95 CI 2.6, 8.4
• and 37 of 253 (p214.6, or .146) switched to
  placebo developed another VTE
• 95 CI 10.3, 18.9
• Could this difference be due to chance?
• Is this difference likely to be due to chance?
• Homework What is ARR and its 95CI, the RRR,
  and NNT?

New Engl. J. Med. 2003 348 1425-1434
17
Chi Square/Fisher Exact Tests
(used for categorical outcomes)

• A new treatment for colitis is compared to the
  standard treatment in 245 patients.
• 120 patients are randomized to the new treatment
  and 125 to the standard treatment.
• 90 given the new treatment group go into
  remission (75) and 30 (25) do not.
• 75 given the standard treatment go into remission
  (60) and 50 (40) do not.
• Is this a significant improvement in outcome, or
  to what extent could this have been due to
  chance? Lets vote!

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Step 1 standard 2X2 table

• New Rx a b ab
• Standard Rx c d cd
• a c b d
  abcdntotal
  patients in study

REMIT
NO REMIT
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Enter the data from our study

• New Rx 90(a) 30(b) 120(ab)
• Standard Rx 75(c) 50(d) 125(cd)
• 165 80
  245(abcd)n

REMIT
NO REMIT
(ac)
(bd)
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Calculate chi square (?2) by plugging in numbers
into handheld or online calculator

• ?2 n (?ad-bc?- n/2)2
• (ab)(cd)(ac)(bd)
• ?2 6.264 (p0.0123)

http//www.graphpad.com/quickcalcs/index.cfm
Fisher exact test, p0.0143
21
We could also have calculated the odds ratio for
a remission

• New Rx a90 b30
• Standard Rx c 75 d50
• odds ratio ad/bc
• odds ratio 4,500/ 2,250 2
• But this odds ratio of 2 could have occurred by
  chance.
• We can calculate the 95 CI of the odds ratio to
  see if the CI overlaps 1 or not. If not, it
  favors the new treatment with gt95 confidence.

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95 CI of an odds ratio

• ln 95 CI ln OR ? 1.96? 1/a1/b1/c1/d
• The OR 2.00, and so the ln 2.00 0.693
• Thus ln 95 CI 0.693 ? 0.508 0.185, 1.201.
• To find the CI, we need the antiln of 0.185 and
  of 1.201.
• Antiln 0.185 e.185 1.20 and antiln 1.201
  e1.201 3.32. ? 95 CI 1.20, 3.32.
• Thus, the odds ratio for a remission with the new
  treatment is 2.00 (95 CI 1.20, 3.32).
• As this odds ratio does not cross 1.00, the
  difference is unlikely due to chance and is
  significant at the 0.05 level.

e?2.72
23
Type 1 and 2 Errors
Null Hypothesis no differences in 2 treatments
Reject null hypothesis
Accept null hypothesis
Correct decision (no error)
Error
Correct decision (no error)
Error
Type 1 (?)
Type 2 (?)
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Choosing ? and ?

• ? (or p) is conventionally set at 0.05 (5), the
  chance of a type 1 error if the null hypothesis
  is rejected (? 5)
• Can state plt0.05 or give exact p value (e.g.,
  p0.01)
• ? is often set at 2 to 4 times ? , or 0.10-0.20
  (10-20)-- the chance of making a type 2 error
  if the null hypothesis is accepted
• Power to detect a real difference (and thus to
  reject the null hypothesis of no difference) 1-
  ?
• tiny ?, large power large ?, little power
• If a study is highly powered and the null
  hypothesis is accepted, the chance of there being
  a true difference is quite small.
• If the study is under-powered and the null
  hypothesis is accepted, there is little
  confidence that a true difference has been
  excluded.

25
Sample size in study planning
A new antibiotic is developed for C. difficile.
How many patients would be needed to be included
in a phase 3 trial to be able to show that this
new drug is superior to metronidazole? To answer
this question, we need to know 1. What is the
response rate for metronidazole? P1 2. What
would be a clinically significant and reasonably
predictable improvement (based on phase 1 and 2
studies) with the new drug? P2 3. What should
be the ? (type 1) and the ? (type 2) error of the
study? (Recall The power of the study to detect
a true difference 1- ?.)
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Sample size estimation, contd

• P1 0.75 (metronidazole)
• P2 0.90 (New Rx)
• ? 0.05 (1 in 20)
• ? 0.10 (1 in 10)
• Power 0.90 (9 in 10)
• N1 and N2 158 per group (Fleiss tables)
• If 10 drop out is expected, then 15816174 per
  group
• Analyze data by intent-to-treat and evaluable
  patients

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Other key concepts

• Sensitivity true positives
• 1-Sensitivity false negatives
• Specificity true negatives
• 1-Specificity false positives
• Likelihood ratio is ratio of the truesfalses
• likelihood ratio sensitivity/1-specificity
• i.e., true / false
• - likelihood ratio specificity/1- sensitivity
• i.e., true -/false -

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Using likelihood ratios

• You have a patient with COPD and an acute onset
  of worsening dyspnea. There is no leg swelling or
  leg pain, hemoptysis, previous PE or DVT, or
  malignancy. However, he had knee surgery 2 weeks
  ago. You assess his odds of PE as fairly low,
  perhaps 101 (10 against to 1 for a PE.)
• How would a CT angiogram change the likelihood
  of PE if ? If - ? In other words, how good is
  CTA in diagnosing or excluding a PE in your
  patient?

29
Using likelihood ratios to calculate posttest
odds

• Literature CTA and pulmonary angiogram (gold
  standard) were assessed in 250 patients with
  possible PE. 50 (20) had PE on pulmonary
  angiography. Results
• CTA CTA- Total
• PE on pulm angio 35 15 50
• No PE on pulm angio 2 198 200
• Likelihood ratio (LR) calculation
• CTA sensitivity (true ).70 1-sensitivity
  (false - ).30
• CTA specificity (true - ).99 1-specificity
  (false ).01
• LR of PE if CTA sensitivity/1-specificity
  true/false 70
• -LR of PE if CTA 1-sensitivity/specificity
  true-/false- .33
• Post test odds (if CTA) (pre-test odds)( LR)
• Posttest odds of PE are now (101) (170)
  1070, or 17 (1 against, to 7 for)
• Post test odds (for CTA) (pre-test odds)(-LR)
  
• Posttest odds of PE are now (101)(10.33)
  100.33 or 331 (33 against, to 1 for a PE).
• Annals Internal Medicine 136 286-287, 2002

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What test(s) to use ?

• DIFFERENCES

Data normally distributed? Paired t (each subject
is his/her own control) Unpaired t (group t)
using mean, SD, and n Data not normally
distributed? Continuous variable? Mann Whitney
U test Wilcoxons sign rank test Categorical
variable? Fishers exact Chi Square Multiple
(gt2) Groups Analysis of variance (ANOVA)

• CORRELATIONS
• Normally distributed
• Pearsons test
• Not normally distributed
• Spearmans test

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Other advanced topics to read about(? future
lectures)

• Kaplan-Meier survival curves
• Logistic regression
• Unadjusted vs. adjusted odds ratios
• Stepwise multivariate discriminate analysis
• Cox proportional hazard analysis
• Meta-analysis, which combine single studies
• Receiver operator curves which plot sensitivity,
  or true s (Y axis) vs. 1-specificity, or false
  s (X axis) using different cutoff points

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Free online websites

• http//faculty.Vassar.edu/lowry/VassarStats.html
• http//www.graphpad.com/quickcalcs/index.cfm
• http//elegans.swmed.edu/leon/stats/utest.html

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34
He who produces an atmosphere of fear and
trembling into the studio has no business
teaching in it.

• Constantine S. Stanislavsky
• 1863-1938

35
ARR/ and its 95 CI

• The absolute risk reduction (ARR) is 14.6
  (placebo) minus 5.5 (warfarin), or 9.1 (0.091).
  
• The 95 CI of this ARR 9.1 ? 7.3 or
  1.8, 16.4.
• Thus, the ARR with warfarin is between 1.8 and
  16.4, with 95 confidence.
• This ARR does not overlap zero.

36
NNT and RRR

• Number needed to treat 1/ARR1/.09111
• Relative risk reduction (RRR) ARR /risk with
  placebo..
• RRR 9.1/14.6 62.3
• However, the length of follow up was not
  identical in the 2 groups within the study.
  People followed longer are at higher risk due to
  this factor alone.
• Adjusting RRR for differences in length of follow
  up
• 7.2 DVTs/1,000 pt.-yrs vs. 2.6/1,000 pt.-yrs
• adjusted RRR (7.2-2.6)/7.2 63.8

37
The normal (bell-shaped) distribution
mean
n
Standard deviations (SD) from the mean. 95 of
values are within 2 SD of mean
38
An example Systolic BP in 11 CVA patients in an
ED

• 240
• 170
• 165
• 140
• 135
• 130
• 120
• 120
• 115
• 100
• 95

Range 95-240 mm Hg Median 130 mm Hg Mean 139
mm Hg
39
Variability The standard deviation (SD)

• 240
• 170
• 165
• 140
• 135
• 130
• 120
• 120
• 115
• 100
• 95

• Between-subject variability can be quantitated by
  calculating the SD, assuming a normal
  distribution of BP readings.
• SD ?(?(differences from the mean2 )/n-1)
• SD 41 mm Hg