Statistical Inference

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Statistical Inference

• Hypothesis Testing, Types of Errors, Probability,
  Statistical Power

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Statistical Inference

• A representative sample of the population is
  studied, then an attempt is made to extrapolate
  the conclusions to the population as a whole
• Test the hypothesis, looking for significant
  differences between the treatment and control
  group.
• Look for errors in sampling

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Definitions

• Population- collection of people which have at
  least one characteristic in common
• Sample- a portion of the population
• Parameter- things that characterize the
  population (standard deviation, mean)
• Statistics- tests used to estimate parameters on
  a sample from the population

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Example

• A controlled study was performed in 61 patients
  with severe alcoholic hepatitis to determine the
  effect of prednisone on their survival rate. The
  mean cumulative survival rate at 2 months was 88
  in the prednisone group compared to 45 in
  patients receiving placebo.
• What is the statistic to be determined in the
  study?

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Hypothesis Testing

• Null Hypothesis the hypothesis which is directly
  tested using statistical tests. Only when the
  null hypothesis is rejected, can the alternative
  hypothesis be accepted.
• Alternative hypothesis Statement which indicates
  that there is a difference between the groups
  studied. One-tailed or two-tailed.
• Estimation- process of using data from sample and
  make conclusions about population parameters.

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Buspirone Example

• A studied stated, The objective was to determine
  if Buspirone will reduce the withdrawal symptoms
  associated with smoking cessation.
• Is this a 1-tailed or 2-tailed hypothesis?

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Hypothesis Testing-- Errors

• Sources of error in a study
• inappropriate selection of sample
• measurement error
• improper assignment of patients to study groups
• random error due to unforeseen factors.

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Types of Errors

• Type I error- rejecting the null hypothesis even
  though it is in fact true. ( You falsely conclude
  that a significant difference exists between
  groups when one really doesnt exist.
• Type II error- failing to reject (accepts) the
  null hypothesis, when in fact it is false.
  (Falsely conclude that no signif. difference
  exists when one really does exist.

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Type I and II Error Example

• A double blind placebo controlled study was
  performed to determine the efficacy of Captopril
  in delaying the progression of congestive heart
  failure. No significant difference was found
  between the two groups in terms of sudden death
  (13 in the Captopril group, and 11 in the
  placebo group).
• What is the null hypothesis?
• If there was an error in the findings from this
  study, which type of error might have been
  responsible for it?

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Alpha

• Alpha- the probability of making a Type I error.
  Set at 0.05 as the cut-off for the amount of Type
  I error one is willing to accept in a study. Also
  called Level of Significance
• Alpha of 0.05-- one is willing to accept a Type I
  error of 5/100 or 1 time out of 20.

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Beta

• Beta- The probability of making a Type II error.A
  beta of less than 0.2 is usually accepted as the
  maximum amount of Type II error one is willing to
  accept.
• Beta is important in calculating the statistical
  power of a study.

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Reducing Type I and Type II Errors

• 1. Make the acceptable Type I error smaller by
  reducing the alpha level (level of significance)
  to less than 0.05.
• 2. Increase the sample size to reduce the
  possibility of Type II error
• The larger the sample size, the more likely it is
  that the study sample represents the population
  and the less chance of sampling error.

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Probability Values

• Usually reported as plt or pgt or p some
  number. Some number is the alpha level or
  probability of a Type I error.
• If plt 0.05, it means that there is less than a 1
  out of 20 likelihood that the difference in
  measured parameters would have resulted from
  chance alone. So the difference is said to be
  statistically significant and due to treatment
  effect.

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Probability Values...

• pgt0.05 means that a difference as large as the
  observed difference might arise from chance and
  the author will not accept this degree of chance.
• It is important that not significant means that
  a difference is not proved it doesnt mean
  that there is absolutely no difference.

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Probability Example

• A study was performed which compared the efficacy
  of Piroxicam and Diflunisal for the treatment of
  osteoarthritis. One group received Piroxicam and
  the other received Diflunisal. After 1 month of
  therapy, the patients who received Piroxicam had
  significant less pain (plt0.002) than the
  Diflunisal group.

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Probability values continued...

• If p 0.04 What does this mean?
• If statistical tests were performed on 25
  different variables, then through chance alone,
  you would expect at least 1 of the tests to
  produce a positive result through a Type I error.
• 0.04 4/100 1/25.

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Probability

• Probability values and CIs compliment each
  other. Example The cure rate was 86 (95
  CI 76-96) for the treatment group as compared to
  the placebo group (plt0.05)
• The 95 CI for the mean of 86 represent the
  likelihood that the true population cure rate
  fall within these ranges (76-96). The p-value
  shows the chance of a Type I error is very small,
  so the results are significant and not due to
  chance.

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Type 1, Type II error Example

• A distribution of the means from repeated samples
  from a group of patients ( group A) given
  Propranolol to reduce BP showed a mean DBP value
  of 73 mmHg (plt0.05). The reduction in group B
  patients given Atenolol showed a mean value of
  75 mmHg (plt0.2).
• 73 mmHg (95CI 69-77)
• 75 mmHG (95CI 65-95)

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Statistical Power

• Power means the ability of a statistical test to
  detect a significant difference between samples
  if the difference actually exists.
• Defined as 1-beta. Since beta is the probability
  of making a Type II error (accepting null
  hypothesis as true when it is really false), then
  1-beta (power) means the probability of NOT
  making a Type II error.

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Statistical Power

• Beta is usually 0.2, then power 1-0.2 0.8
  (ie. 80 or greater).
• Power is determined prior to study.
• Degree of Power depends on
• Sample size -- the larger the sample size, the
  greater the power of statistical tests
• Alpha level- larger the alpha level, smaller the
  beta
• Effect size- refers to difference between
  populations which one would detect as signif.

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Effect Size

• Suppose you are measuring the effects of
  Lovastatin and Pravastatin on Cholesterol levels.
  You would like to detect a cholesterol level drop
  of 20 points as significant. The 20 points would
  be the effect size. The alpha level is set at
  0.01.

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Statistical Power

• Hypothesis testing with the probability values
  provide a yes or no answer concerning whether
  differences exist between the population studied.
  The p value itself doesnt say anything about
  the size of the difference between means.

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Probability

• Example The authors state the drop in
  cholesterol levels was highly significant (plt
  0.001)
• The highly significant is not appropriate. The
  p value only tells you that it is very unlikely
  that the drop in cholesterol levels was due to
  chance, it does NOT tell you the magnitude of the
  difference.

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Example

• A study demonstrated a correlation between
  alcohol consumption and elevated blood pressure.
  The amount of alcohol consumption was compared
  with blood pressure readings. A difference in
  average BP was found between patients who drank
  small amts of alcohol and those with high intake.
  Those who drank more had higher blood pressure.
  This finding was reported as statistically
  significant with a p-value of 0.0001.
• What does this value mean to us?
• It means that chance is an unlikely explanation
  for the results. The extremely small p-value is
  due to the size of the sample 2100
• Does this statistic alone mean that alcohol
  consumption is a cause of hypertension?

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Methadone Example

• Buprenorphine tx was significantly better than
  methadone 20 mg/day (plt0.001) and methadone
  60mg/day was better than methadone 20mg/day
  (plt0.04).
• Would you conclude that the difference between
  buprenorphine and methadone 20mg/d was more
  highly significant than the difference between
  the two methadone txs?

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Statistical Testing

• P values do not describe the size of differences.
• Statistical significance does not imply clinical
  significance
• Consider both statistical significance and
  clinical significance when evaluating the true
  usefulness of the study results and drug.

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Questions to answer before statistical test is
chosen

• 1. What is the main study hypothesis? Is is
  one-tailed or two-tailed?
• 2. Are the data independent or paired/matched?
• 3. What type of data are being measured?
• Scale level of measurement

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One tailed vs. Two tailed tests
0.025 tail (alpha0.05) 0.025 tail
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One Tailed vs. Two Tailed Tests

• The use of a 1-tailed test is appropriate only
  when the authors have clearly stated a 1-tailed
  hypothesis direction.
• Example A study of the efficacy of zinc lozenges
  in treating the common cold stated, We
  determined the duration of cold symptoms in zinc
  and placebo treated patients. A 2-sided p value
  of 0.05 was used in the hypothesis testing.

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Parametric Tests

• 3 types of research questions asked
• 1. Are there differences between or among groups?
• 2. What are the associations among groups?
• 3. What conclusions and predictions can be made
  as a result of the data collected?
• Statistical tests are used to answer these
  questions (parametric and non-parametric)

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Parametric Tests

• To use the data is assumed to follow a normal
  distribution (or near normal)
• Used for continuous level data measurements
  (ratio or interval data)
• More powerful than non-parametric tests. They are
  better able to detect existing differences
  between or among groups

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Non Parametric Tests

• Only used when Parametric tests cant be used.
• Used for Non-continuous data (nominal or ordinal
  level data)
• Used when data are not normally distributed or
  they deviate from normal distribution
  substantially

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Parametric Test T-test or Students T-test

• Sample data is normally (or near normally)
  distributed.
• The variances of the sample populations data are
  nearly equal
• The measurements within the sample population are
  independent of each other.
• Data from only 2 groups are being compared.

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Types of T-tests

• Non-paired t-test, Students t-test-- used when
  different control and experiment groups are used.
• Paired t-test- used when measurements are taken
  in the same group of patients (before and after)
• On either type alpha level predetermined, a
  number called a t-value is calculated using a
  table of t-values determine the p value

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T-tests

• One tailed t-test if a direction of difference
  is postulated
• Two tailed t-test if no direction of difference
  is postulated.

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T-test Example

• A study was performed to determine if Pravastatin
  is more effective than placebo in the lowering of
  triglycerides. 50 patients with moderately
  elevated triglycerides were randomly selected to
  receive Pravastatin and 50 patients were give
  placebo. A t-test was performed using the mean
  triglyceride level in both groups at the end of
  12 weeks. The difference in triglyceride levels
  from baseline was statistically significant (plt
  0.05)

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Example

• A study to see the effect of Amiodarone on serum
  digoxin concentrations was done. 30 patients with
  heart failure who were taking digoxin and had an
  indication for amiodarone were enrolled if their
  serum digoxin concentration was not more than 1.5
  ng/ml. Each subject had a serum digoxin
  concentration measured before starting Amiodarone
  and after 3 months.
• What type of t-test?
• Paired or non-paired
• one-tailed or two-tailed?

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Calculating the t-value

• Suppose a new drug is being tested to see if it
  will decrease arterial pressure in people with
  hypertension.
• 2 sample groups, alpha level pre-set, 1-tail.
• Data collected, descriptive statistics applied
• t-value computed from equations
• Table of critical t-values consulted

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t-test calculation
XT - XC t varT
varC nT nC X
mean, T treatment, C control var variance , n
sample number
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Multiple T-tests

• Multiple t-tests are NOT appropriate when
  comparing more than 2 groups.
• The probability of making a type I error
  increases as the number of tests are performed.

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Analysis of Variance (ANOVA)

• When comparing differences between 3 or more
  groups.
• More powerful than using multiple t-tests
• Type 1 error (alpha level) stays constant
  regardless of the number of groups in the design.
• Examining the variability both between and within
  the study groups

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Analysis of Variance (ANOVA)

• Assumptions to use ANOVA
• data is continuous level (interval or ratio) and
  near normally distributed.
• The variance of the populations from which the
  samples were drawn are nearly equal.
• The observations or measurements within a
  population or sample are independent of each
  other.

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F-Ratio

• ANOVA involves calculation of a F-ratio, which is
  compared with a critical F-ratio in a table.
• F-Ratio Between groups variance
• Within groups variance
• Used to answer question Is the variability
  between groups large enough in comparison to
  variability within groups to say that the groups
  differ.

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F-Ratio

• If the between groups variance and the within
  groups variance are about equal, then the value
  of the F-ratio will be small and the null
  hypothesis will be assumed to be true.
• If the F-ratio is large (between groups effect is
  greater than any effect within groups), the more
  likely it is that real differences in drug
  treatments will be found.
• F-ratios reported as plt0.05, etc.

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Analysis of Variance

• One-way ANOVA an independent variable in an
  ANOVA is called a factor. Each factor may have
  several levels such as a treatment factor with
  several doses. A study of ONE factor is called a
  one-way ANOVA.
• A study of two different doses of a drug at 3
  different times has two factors (drug/time). Use
  a two-way ANOVA

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Analysis of Variance

• Repeated Measures ANOVA used when measurements
  are repeated in the study groups over time.
• Example The antihypertensive effect of a new
  drug (Drug X) is being compared to Clonidine. BP
  measurements for both drug groups are taken after
  1 week, 2 weeks and 1 month.

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ANOVA Example

• Study to looked at efficacy of Desipramine,
  Amitriptyline and placebo in pain relief for
  peripheral neuropathy patients. The investigators
  also studied whether depressed patients responded
  any differently compared to non-depressed
  patients using Desipramine and Amitriptyline.
• What type of ANOVA would be indicated?

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Analysis of Variance

• After ANOVA, researcher states either that no
  difference exists among groups or that a
  difference exists somewhere, but this test does
  not indicate which of the groups differs from the
  others. Other tests are then applied to find the
  specific group responsible for the differences.
  (Least Signif.Difference (LSD) test, Neuman-Keuls
  test, Dunnet test, Tukey, Dunn and Scheffes
  procedures, Bonferroni Procedure

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Other Tests- Multiple Comparison Tests

• Dunnetts test- only when comparing several tx
  group means with one control group mean.
• Tukey, Scheffes- used when large number of
  comparisons are made
• Bonferroni- Correction to the use of multiple
  t-tests, which takes into account the number of
  comparisons being made

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Non-Parametric Statistical Tests

• Used when
• the data distribution diverges from expected
  normal characteristics
• when the scale level of measurement of data is
  not continuous (ie. Nominal or ordinal)
• when the data is continuous but does not meet the
  criteria for parametric tests
• Nonparametric tests lt powerful than parametric

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Nominal Data Tests

• Chi Square test (x2) used for frequencies and
  proportions and distributions.
• Matrix is made w/ rows and columns into cells.
  Calculations are compared between observed values
  and expected values. The greater the difference
  between tx, the larger X2 will be. From
  statistical table it is determined if the value
  is large enough for statistical significance.
  Reported as plt0.05

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Chi Square 2 x 2 table
Outcome Cimetidine Ranitidine Total GI
bleed 22 (18.5) 15 (18.5) 37 No bleed 18
(21.5) 25 (21.5) 43 ( ) expected frequencies if
no differences between groups. E11 row 1 total X
column 1 total N (total in all cells) E11
(2215) X (2218) 37 X 40 18.5 37
43 80
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Calculating the x2

• 1. Subtract each expected number from each
  observed number in each cell. (O-E)
• Square the difference. (O-E)2
• Divide the squares obtained for each cell of the
  table by the expected number for that cell
  (O-E)2
• E
• X2 the sum of all cells numbers
• Use X2 and df to get p-value from table.

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Chi Square Table
Distribution of Probability d.f. 0.5 0.10
0.05 0.02 0.01 0.00l 1 0.455 2.706
3.841 5.412 6.635 10.827 2 1.386 4.605
5.991 7.824 9.210 13.815 3 2.366 6.251
7.815 9.837 11.345 16.268 4 3.357 7.779
9.488 11.668 13.277 18.465 5 4.351 9.236
11.070 13.388 15.086 20.517 degrees of freedom
columns minus one X rows minus one
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Chi Square Example
Example A study was done to determine efficacy
of Cimetidine, Ranitidine and Famotidine for
preventing GI bleeding in critically ill
patients. Each drug was given to 40 pts. The
number of pts. experiencing a GI bleed were 22,
15 and 16 respectively. Outcome Cimetidine Ranitid
ine Famotidine GI bleed 22 15 16 No
bleed 18 25 24 How many total cells would this
table contain?
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Chi Square Test (x2) For Nominal Data

• Assumptions for use with Chi Square test
• total number of observations must be greater than
  20
• No more than 20 of cells have expected frequency
  of less than 5.
• For a 2 X 2 table, no cell has expected freq. lt5
• For a 2 X 3 table, 1 cell can have exp.freq lt5
• Samples must be independent from each other

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Yates Correction Factor

• Applied to 2 x 2 chi-square tables and relatively
  small sample size (lt40)
• Also called Continuity Correction
• Advantage reduces the risk of a type I error.
  But when type I error risk reduced, type II
  error risk is increased. If type II error risk
  increased, power of test decreases.

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Non Parametric Tests For Nominal Data

• Fishers Exact Test
• only used for 2 X 2 tables.
• Used for Chi Square when the total of observ.
  Is lt20, but each cell still has an expected freq.
  of not less than 5 OR total of observ. Is gt20,
  but one of the cells has an expected frequency
  value of less than 5.

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Fishers Exact Test Example
Example The efficacy of heparin and low dose
warfarin for the prevention of deep vein
thrombosis (DVT) was determined. 20 patients
received heparin and 25 received warfarin. DVT
occurred in 3 patients in the heparin group and 5
patients in the warfarin group. Outcome Heparin Wa
rfarin Total DVT 3 (3.6) 5 (4.4) 8 No
DVT 17 20 37
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Mantel-Haenszel Procedure

• Correction test used when you want to adjust for
  extraneous independent variables that could
  influence the outcome of a test.
• When a Fisher Exact test hasnt found statistical
  significance, sometimes when combined with the
  Mantel-Haenszel one can obtain statistical
  significance.

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Non Parametric Tests For Nominal Data- McNemars
Test

• McNemars Test
• Level of Data is nominal
• Used when study samples are not independent
• Can also be applied to prospective cohort studies
  or case-control studies in which each member is
  paired with a control.

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McNemar Test Example
The efficacy of two anti-nausea drugs was
determined in 20 patients receiving chemotherapy
every month. During the first month of
chemotherapy, the patients received drug l and
the presence or absence of nausea was determined.
The next month, the patients received Drug 2 and
the presence or absence of nausea was again
assessed. The data are nominal (presence or
absence) The same group of patients were
evaluated (paired sample- not independent
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Non-Parametric Tests for Ordinal Data

• Mann-Whitney U test (Single comparison)--
  comparison between only 2 independent groups
  (Drug A group vs. Drug B group.
• Similar to the non-paired t-test in parametric
  tests.
• Used for ordinal data, or for continuous data
  only when its not normally distributed, or when
  assumptions for t-test are violated.
• Can identify where signif. differences exist
  between pairs

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Non Parametric Tests for Ordinal Data

• Wilcoxon Signed Rank test
• Used for ordinal data or interval level data when
  other t-test assumptions arent met.
• Used when 2 sample groups are NOT independent
  (when same patients receive the different tx and
  serve as their own controls)
• Counterpart to paired t-test.

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Non Parametric Tests for Ordinal Data (multiple
comparison test)

• Kruskal-Wallis test (counterpart to one-way
  ANOVA).Used for
• comparison of 3 or more samples.
• Used on ordinal level data or continuous when
  there might be violations for ANOVA
• Analysis of one independent variable
• Does not tell where significant difference is.
• Use Mann-Whitney U test to locate differences.

66
Non Parametric Tests for Ordinal Data (Multiple
comparison tests)

• Friedman Test- For comparison among more than 3
  groups.
• counterpart to repeated measures ANOVA
• used for ordinal level data or continuous level
  data when other assumptions for ANOVA are
  violated.

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Correlation Analysis

• The statistical method to describe the strength
  and direction of the association of 2 or more
  variables.
• Example Is there a relationship between
  cigarette smoking and atherosclerotic heart
  disease?

AHD
cigarettes/day
68
If two variables are correlated are they causally
related?

• Do not confuse correlation with causation.
• Correlation shows that the two variables are
  associated. There may be a third variable or a
  confounding variable that is related to both of
  them.
• Example Monthly deaths by drowning and monthly
  sales of ice-cream are positively correlated, but
  does one cause the other?

69
Correlation between 2 variables

• In an efficacy study, you may want to know the
  relationship between drug dosage and the degree
  of response.
• If the 2 variables are correlated, the values for
  one variable will vary depending upon the values
  of the other.
• The number calculated from the analysis is called
  a correlation coefficient r

70
Correlation Coefficient r

• r only ranges in value from 1 to -1.
• If r0, there is NO linear association between
  variables.(data points scattered)
• As r approaches 1, the association becomes
  stronger (more linear) in a postive direction.
  One variable increases, the other variable
  increases also.
• As r approaches -1, assoc. becomes stronger in
  a negative direction.

71
Correlation Coefficient r
R 0.86
R 0.1
R - 0.92
72
Correlation Analysis Tests

• Pearson (Pearson product-moment) r
• Used to describe the strength and direction of
  the relationship between 2 variables that are
• continuous level data (interval/ratio)
• follow a normal distribution pattern
• have a linear relationship

73
Correlation Analysis Tests

• Spearman (Spearman rank order) r
• This is used to describe the strength and
  direction of relationship between 2 variables
• in which at least 1 variable is ordinal level
  data
• or which are continuous but do not follow normal
  distribution patterns.

74
Correlation Coefficient r

• An r from 0 to /- 0.25 indicates no or only
  slight linear relationship between var.
• An r from /- 0.25 to /- 0.75 indicates linear
  relationship which is weak to fairly strong
• An r from /- 0.75 - /- 1 indicates a strong
  to very strong linear relationship between
  variables.

75
Correlation Coefficient r

• For a given r value, you can predict how much
  of the variability in one measurement can be
  accounted for by the presence of the other.
• If you square r (r2), the resulting number is
  used as a percentage estimate of this variability

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Example

• The correlation of the blood pressure lowering
  effect of Atenolol with patients renal function
  status was reported as r 0.4
• What percent of the variability in the patients
  BP response to Atenolol can be explained by
  differences in renal function?
• r2 0.42 0.16 16 can be explained by
  differences in renal function.

77
Correlation Coefficient (r)

• There is no direct proportional relationship
  between different r values. (0.4 is not half the
  strength of 0.8)
• Outlying data points will effect the strength of
  the linear relationship and will lower or raise
  the r value, depending on where the outliers are.

78
Statistical Significance of Correlation

• p values will be reported along with correlation
  coefficients so you can determine if the r
  value is statistically significant. But remember
• The larger the number of data points in a
  correlation analysis, the more likely it is that
  even a small r value will be statistically
  signif.
• A statistically signif. r value doesnt mean
  clinical significance.

79
Correlation Example
1.A study examines the relationship between a
patients height and the blood pressure effect of
Clonidine. 500 patients were included and a
correlation coefficient of r 0.17 was
determined. (p0.026) Can you conclude a
substantial relationship exists between height
and Clonidines BP effect? 2. How about with
an r 0.85 and a p 0.063? 3. Would an r 0.85
represent a stronger degree of correlation
between 2 variables than an r - 0.85?
80
Example Problem

• You collect data on the ages and weights of 100
  persons- both children and adults- who were
  attending picnic. If you calculate the
  correlation coefficient between the ages and the
  weights of these 100 persons,which of the
  following would most likely be the correlation
  you would find?
• A. -0.90 B. 1.00 C. 0.02 D. 0.50

81
Regression

• Linear Regression (simple and multivariate)
• Association between one variable as a function of
  the other variable
• Pearson Product moment testnormal distr,
  continuous data
• Spearman Rank test ordinal data
• Logistic Regression
• Stepwise multiple regression that involve
  manipulating multiple variables simultaneously to
  determine which best predicts the outcome.
• Uses nominal data and is used to compute odds
  ratio

82
Regression when two variables are related

• Correlation describes the strength of association
  between two variables and is completely
  symmetrical.
• If two variables are related means that when one
  changes by a certain amount the other changes by
  a certain amount also.
• The relationship of how they are related is
  called the regression line and is based on the
  math computation of a straight line
• y a Bx (remember y 2x b )

83
Regression line
84
Multiple Regression Analysis(Multivariate
Analysis)

• Testing multiple variables
• Testing whether confounding variables influence
  outcomes
• Example relationship between age and blood
  pressure, with sodium intake as a possible
  confounder variable.

85
Analysis of Covariance (ANCOVA)

• Combination of ANOVA (discrete independent
  variables) and regression (quantitative
  independent variables)
• Discrete variable is treatment or class
  variable
• Quantitative independent variable is covariate,
  continuous or independent variable.
• This test will show the interactions and
  influences of the covariate on the class variable

86
Survival analysis

• Collection of statistical procedures for analysis
  of data in which the variable of interest is
  time until an event occurs.
• Often seen as hazard ratio when the event of
  interest is death, the hazard association with
  a particular moment in time is the probability of
  death at that moment or survival until that moment

87
Survival Analysis cont..

• Kaplan-Meier curves show the probability of being
  alive at any specified future time.
• Kaplan-Meier curves for more than 2 groups can be
  evaluated to determine whether the curves are
  statistically different by using the log rank
  test (which is a large sample chi square test)

88
Survival Analysis cont

• Cox proportional hazard models
• Non-parametric test that provides an estimate of
  a hazard function. The ratio of the hazards for 2
  individuals, one with and one without a risk
  factor, given equal covariates can be estimated.