BASIC STATISTICS

1
BASIC STATISTICS

• Alan J. Chaput, BScPhm, PharmD, MD, MSc (Epid),
  FRCPC
• June 28, 2007

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Descriptive statistics
3
Types of data

• Categorical
• Places individuals into one of several categories
• Quantitative (numerical)
• Numerical values for which arithmetic operations
  such as adding and averaging make sense

4
How would you categorize these variables?

• Height
• Hours of sleep
• Hair color
• Eye color
• Ever taken stats course
• Heart rate
• 7-point Likert scale

5
Distribution of a variable

• Tells us values a variable takes and how often it
  takes them
• Start data analyses by exploring distributions of
  single variables with a graph
• Later, move on to studying relationships between
  variables

6
Displaying distributions

• Categorical
• Pie charts
• Bar graphs
• Quantitative
• Histograms
• Stem-and-leaf plots
• Boxplots

7
Graphs vs. tables

• Graphs
• A pictures worth
• Can show ALL the data points
• Visual impact of data (presentations)
• Tables
• More efficient (usually)
• Show actual values (more precision)
• Easier to produce (historically)

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Categorical VariablesExample U.S. Solid Waste
(2000)
Pie Chart
Difficult to do by hand, use computer program
(e.g., Excel) for production if necessary.
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Categorical Variables Example U.S. Solid Waste
(2000)
Bar Graph
Notes 1) Bars do not touch (categorical data)
2) Can plot counts or percents
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Quantitative variables - histograms

• Divide data into class intervals of equal width
• Count how many observations fall in each interval
• Draw histogram

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Weight Data Histogram
Number of students
Weight
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Interpreting histograms

• Shape
• Symmetric (bell, other)
• Asymmetric (right-tail, left-tail)
• Unimodal, bimodal (mode a high point)
• Centre (find middle position)
• Count observations (n)
• Find middle position (n 1) / 2
• Find middle value the value that has the middle
  position
• Spread (from low to high)
• Outliers (values outside the regular pattern)

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Interpreting histogramIllustrative example
State population, Hispanic (Fig 1.3)

• Shape asymmetrical w/ right tail
• Center
• n 50 states
• Middle position (50 1) 2 25.5
• Middle value is in the first category, so is
  between 0 and 5
• Spread From 0.7 (W. Virginia) to 42.1 (New
  Mexico)
• Outlier New Mexico

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Interpreting histogramIllustrative example Fig
1.4 in text

• Shape symmetrical bell
• Center
• n 100
• Middle position 50.5
• Middle value around 7
• Spread From 2 to 12
• Outlier 12 (maybe)

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Stem-and-leaf plots

• For quantitative variables
• Separate each value into a stem value (first part
  of the number) and leaf value (the remaining part
  of the number)
• Create a stem axis
• Write each leaf to the right of its stem value
• Place leaves in rank order
• Interpretation like a histogram on its side

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Weight DataStemplot(Stem Leaf Plot)
10 0166 11 009 12 0034578 13 00359 14 08 15
00257 16 555 17 000255 18 000055567 19 245 20
3 21 025 22 0 23 24 25 26 0 (10)
Key 203 means203 pounds Stems 10sLeaves
1s
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Interpretation like a histogram on its side
100166 11009 120034578 1300359 1408 1500257
16555 17000255 18000055567 19245 203 21025
220 23 24 25 260 (10)

• Shape positive skew
• Center
• n 53
• middle position (531)/2 27
• middle value 165
• Spread from 100 to 260
• Outlier 260

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Boxplot

• Central box spans Q1 to Q3
• A line in the box marks the median
• Lines extend from the box out to the minimum and
  maximum

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Weight Data Boxplot
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Boxplots are esp. useful for comparing groups
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Numerical summaries

• Centre
• Median
• Mean
• Spread
• Quartiles (and IQR)
• Standard deviation (and variance)

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Mean (Arithmetic Average)

• Traditional measure of center
• Notation (xbar)
• Sum the values and divide by the sample size (n)

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Median

• Half the ordered values are less than or equal to
  the median (and half are greater)
• If n is odd, the median is the middle ordered
  value
• If n is even, the median is the average of the
  two middle ordered values

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Comparing the mean and median

• Mean median when data are symmetrical
• Mean ? median when data are skewed

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Spread variability

• Variability the amount the values spread above
  and below the centre
• Can be measured in several ways
• Range
• Quartiles and inter-quartile range
• Variance and standard deviation

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Range

• Range maximum minimum
• The range is NOT a reliable measure of spread

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Quartiles

• The quartiles are the 3 numbers that divide the
  ordered data into 4 equally sized groups
• Q1 has 25 of the data below it
• Q2 has 50 of the data below it (median)
• Q3 has 75 of the data below it

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Obtaining the quartiles

• Order the data
• Find the median (this is Q2)
• Look at the lower half of the data (those below
  the median)
• The median of this lower half Q1
• Look at the upper half of the data (those above
  the median)
• The median of this upper half Q3

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5 number summary

• Minimum
• Q1
• Median (Q2)
• Q3
• Maximum
• Note
• IQR Q3 Q1
• IQR gives spread of middle 50 of data

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Variances and standard deviation

• The most common measures of spread
• Based on deviations around the mean
• Each data value has a deviation

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Variance Formula
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Standard Deviation Square root of the variance
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Choosing summary statistics

• Use the mean and standard deviation for
  reasonably symmetric distributions that are free
  of outliers
• Use the median and IQR (or 5-point summary) when
  data are skewed or when outliers are present

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The normal distribution
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Who is this??
36
Mathematical formula of a normal curve
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Normal curves

• Bell-shaped
• Not too steep, not too fat
• Defined by means standard deviations

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Normal Curves

• The mean and standard deviation computed from
  actual observations (data) are denoted by and
  s, respectively.

• The mean and standard deviation of the
  distribution represented by the density curve are
  denoted by µ (mu) and ? (sigma), respectively.

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Bell-Shaped CurveThe Normal Distribution
Standard deviation
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The Normal Distribution

• Mean µ defines the center of the curve
• Standard deviation ? defines the spread
• Notation is N(µ,?).

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68-95-99.7 Rule forAny Normal Curve

• 68 of the observations fall within one standard
  deviation of the mean
• 95 of the observations fall within two standard
  deviations of the mean
• 99.7 of the observations fall within three
  standard deviations of the mean

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68-95-99.7 Rule forAny Normal Curve
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Standard Normal (Z) Distribution

• The Standard Normal distribution has mean 0 and
  standard deviation 1
• We call this a Z distribution ZN(0,1)
• Any Normal variable x can be turned into a Z
  variable (standardized) by subtracting µ and
  dividing by s

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Standard Normal Table
45
Statistical tests of normality

• Kolmogorov-Smirnoff test
• Anderson-Darling test
• Shapiro-Wilk test
• DAgostino-Pearson omnibus test

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Idea of probability

• Probability is the science of chance behavior
• Chance behavior is unpredictable in the short
  run, but has a predictable pattern in the long
  run
• A phenomenon is random if individual outcomes are
  uncertain but there is a predictable distribution
  of outcomes in a large number of repetitions.

The probability of any outcome of a random
phenomenon can be defined as the proportion of
times the outcome would occur in a very long
series of repetitions.
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How probabilities behave
Eventually, the proportion of heads in fair coin
tosses approaches 0.5
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Recall the Normal curve

• We use the Normal density curve to determine
  probabilities

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Normal probability distribution
individuals with X such that x1 lt X lt x2
The shaded area under the density curve shows the
proportion, or percent, of individuals in the
population with values of X between x1 and x2.
Because the probability of drawing one individual
at random depends on the frequency of this type
of individual in the population, the probability
is also the shaded area under the curve.
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Normal probability distribution
A variable whose value is a number resulting from
a random process is a random variable. The
probability distribution of many random variables
is the normal distribution. It shows what values
the random variable can take and is used to
assign probabilities to those values.
Example Probability distribution of womens
heights. Here, since we chose a woman randomly,
her height, X, is a random variable.
To calculate probabilities with the normal
distribution, we will standardize the random
variable
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Mens Height Example (NHANES, 1980)

• What proportion of men are less than 68 inches
  tall?

-0.71 0 (standardized values)
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Standardized Scores

• How many standard deviations is 68 from µ on
  XN(70,2.8)?
• z (x µ) / s
• (68 - 70) / 2.8 -0.71
• The value 68 is 0.71 standard deviations below
  the mean 70

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Table AStandard Normal Probabilities
.01
?0.7
.2389
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Mens Height Example (NHANES, 1980)

• What proportion of men are greater than 68 inches
  tall?
• Area under curve sums to 1, so Pr(X gt x) 1
  Pr(X lt x), as shown below

1?.2389 .7611
.2389
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Reminder standardizing N (m,s)
We standardize Normal data by calculating
z-scores.
Any N(µ,s) can be standardized to a N(0,1).
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Distribution of womens heights
N (µ, ?) N (64.5, 2.5)
Example What's the proportion of women with a
height between 57" and 72"? Thats within 3
standard deviations s of the mean m, thus that
proportion is roughly 99.7.
Since about 99.7 of all women have heights
between 57" and 72", the chance of picking one
woman at random with a height in that range is
also about 99.7.
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What is the probability, if we pick one woman at
random, that her height will be some value X?
For instance, between 68 and 70 P(68 lt X lt
70)? Because the woman is selected at random, X
is a random variable.
N(µ, s) N(64.5, 2.5)
0.9192
0.9861
The area under the curve for the interval 68" to
70" is 0.9861 - 0.9192 0.0669. Thus, the
probability that a randomly chosen woman falls
into this range is 6.69
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Odds and probability
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Standard 2x2 table
Outcome/Disease
Exposure/Treatment
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Calculations from 2x2 table

• RR a/(ab) / c/(cd)
• RRR c/(cd) a/(ab) / c/(cd)
• ARR c/(cd) a/(ab)
• NNT 1/ARR
• OR a/b / c/d ad/cb

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Why use odds ratios?

• Perfectly good measure of association
• Can be estimated from a case-control study (RR
  cannot)
• Offers advantages in meta-analysis
• Easier to model than RD or RR
• As the risk falls, the odds and risk come closer
  together for low event rates, the OR and RR are
  very close

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Odds and probability/risk

• Odds are an alternative way of describing the
  chance of an event
• Odds Prob / 1- Prob
• Rearranging
• Prob Odds / 1 Odds

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Risk and odds
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Producing data sampling
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Inference!

• We often want answers to questions about a large
  group of individuals -- this is the population
• We seldom study the entire population, but
  instead select a subset of the population -- this
  is the sample
• We use inferential techniques to make conclusions
  about the population based on the data in the
  sample

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Two types of studies

• Observational individuals are studied without
  intervention
• E.g., case-control and cohort studies
• Experimental studies the investigator assigns
  an explanatory factor to some subjects but not to
  others
• E.g., clinical trial

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Observations vs. Experiments

• Both types of studies may be used to learn about
  relationships between variables
• Experimental studies are better suited for
  determining cause-and-effect because they can
  deal with confounding variables via randomization

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Sample Quality

• To do a good study, you need a good sample
• Poor quality samples produce misleading results
• Study should be designed to generate high quality
  data that can then be used to infer population
  characteristics

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Examples of Poor Quality Sampling Designs

• Voluntary response sampling
• Allows individuals to choose to be in the study
• e.g., Call-in polls (pp. 1789 in text)
• Convenience sampling
• Individuals that are easiest to reach are
  selected
• e.g., Interviewing at the mall (p. 179)
• These techniques favor certain outcomes and
  cannot be trusted to reveal population
  characteristics (sampling bias)

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Simple Random Sample (SRS)

• To avoid biased sampling, use impersonal chance
  mechanisms as the basis for selection
• Simple Random Sample (SRS)
• (1) Each individual in population has the same
  chance of being selected
• (2) Every possible sample has an equal chance to
  be chosen

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Methods for selecting SRSs

• Physical, e.g., pick numbers from a hat
• Computerized random number generators
• Use a table of random digits

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Picking SRS (Illustration)

• Suppose 30 individuals labeled 01 30 and we
  want to select two at random
• Random digit table
• Select a row in table at random
• Break digits into couples
• 68417 35013 15529
• First two individuals in the sample are 13 and
  15

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Producing data experiments
74
Experimentation

• In an experiment, the investigator exposes the
  explanatory factor to some individuals but not to
  others. The investigator then measures the effect
  on the response variable.
• In an observational (non-experimental) study,
  individuals are studied without imposition of
  intervention, creating greater opportunity for
  confounding

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Comparison
Comparison is first principle of experimentation

• The effects of treatment can be judged only in
  relation to what would happen in its absence (all
  other things being equal)
• You cannot assess the effects of a treatment
  without a proper comparison group because
• Many factors contribute to a response
• Conditions change on their own over time
• People are open to suggestion (Placebo effect)
• Observation changes behavior (Hawthorne effect)

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Randomization
Randomization is the second principle of
experimentation

• Randomization use of chance mechanisms to
  assign treatments
• Randomization balances confounding variables
  among treatments groups

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Blinding
Blinding is the third principle of experimentation

• Blinding assessment of the response in subjects
  is made without knowledge of which treatment they
  are receiving
• Single blinding subjects are unaware of
  treatment group
• Double blinding subjects and investigators are
  blinded

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The Logic of Randomization

• Randomization ensures that differences in the
  response are due to either
• Treatment
• Chance in the assignment of treatments
• If an experiment finds a difference among groups,
  we then ask whether this difference is due to the
  treatment or due to chance
• If the observed difference is larger than what
  would be expected just by chance, then we say it
  is statistically significant

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The logic of randomization

• Consider an experiment of weight gain in
  laboratory rats
• Just by luck, some faster-growing rats are going
  to end up in one group or the other
• If we assign many rats to each group, the effects
  of chance will balance out
• Use enough controls to balance out chance
  differences

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Sampling distribution of means
81
Parameters and Statistics

• Parameter a fixed number that describes the
  location or spread of a population (the value of
  parameter NOT known)
• Statistic a number calculated from data in the
  sample (the value of statistics IS known after it
  is calculated)
• Sampling variability different samples or
  experiments from the same population yield
  different values of the same statistic

82
Parameters and statistics

• The mean of a population is denoted µ ? this is
  a parameter
• The mean of a sample is called x-bar ? this is
  a statistic
• Illustration
• Average age of all U of O students (µ) is 26.5
• A SRS of 10 U of O students yields an average age
  (x-bar) of 22.3
• x-bar and µ are related but are not the same
  thing!

83
Law of Large Numbers
The figure to the right demonstrates the law of
large numbers. The average of the first 50 or so
observations is unreliable. As n increases, the
sample mean becomes a better reflection of the
population mean.
84
Sampling distribution of xbar
Key questions What would happen if we took many
samples or did the experiment many times? How
would this affect the statistics calculated from
such samples?
85
Case Study Does This Wine Smell Bad?

• For the variable with ? 25 µg / L, ? 7 µg / L
  with a Normal distribution, suppose we take 1,000
  samples, each of n 10 from this population,
  calculate x-bar from each sample, and plot x-bars
  as a histogram

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The distribution of all x-bars
x-bar is an unbiased estimate of µ
Averages are less variable than individual
observations.
87
Central Limit Theorem
No matter the shape of the population, the
distribution of x-bars will tend to be Normal
when the sample is large.
88
Central Limit Theorem Illustrative example time
to perform activity

• Data time to perform an activity (hours)
• Population NOT Normal (Fig a) with µ 1 s 1
• Fig (b) is for x-bars based on n 2
• Fig (c) is for x-bars based on n 10
• Fig (d) is for x-bars based on n 25
• Distributions become increasingly Normal because
  of the Central Limit Theorem

89
Confidence intervals the basics
90
Statistical Inference

• Two types of statistical inference
• Confidence Intervals
• Tests of Significance

91
Confidence IntervalMean of a Normal Population

• Take an SRS of size n from a Normal population
  with unknown mean m and known standard deviation
  s. A level C confidence interval for m is

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Confidence IntervalMean of a Normal Population
93
How Confidence Intervals Behave

• The margin of error is
• The margin of error gets smaller, resulting in
  more accurate inference,
• when n gets larger
• when z gets smaller (confidence level gets
  smaller)
• when s gets smaller (less variation)

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Interpretation of a confidence interval

• We are (1-a) x 100 confident that the true value
  of µ lies in the interval µL to µH
• If we used the interval several times, then (1-a)
  x 100 of the time, it will cover the true value
  of µ
• The main purpose of a CI is to estimate an
  unknown parameter with an indication of how
  accurate the estimate is and how confident we are
  that the result is accurate

95
Level of Confidence (C)

• Confidence level success rate of method
• e.g., a 95 CI says we got this interval by a
  method that gives correct results 95 of the
  time (next slide)
• Most common levels of confidence are 90, 95,
  and 99

96
Common MISinterpretations of a CI

• The means µ will lie within the interval with
  probability 0.95
• µ is in this interval with probability 0.95
• The mean of a future sample from this population
  will lie in the interval
• 95 of the data will lie in the interval

97
Factors that influence a CI

• The higher the level of confidence, the wider the
  CI
• The larger the variability in the sample, the
  wider the CI
• The larger the sample size, the narrower the CI

98
Tests of significance the basics
99
Recall basics about inference

• Goal to generalize from the sample (statistic)
  to the population (parameter)
• Two forms of inference
• Confidence intervals
• Significance testing
• Both CI and significance testing are based on the
  idea of a sampling distribution

100
Stating Hypotheses

• The goal of this procedure is to quantify the
  evidence against a claim of no difference?
• The claim being tested is called the null
  hypothesis H0
• The null hypothesis is contradicted by the
  alternative hypothesis Ha (which indicates that
  there is difference)
• The test is designed to assess the strength of
  evidence against the null hypothesis.

101
Test hypotheses

• Null H0 m m0
• One sided alternatives
• Ha m gt m0
• Ha m lt m0
• Two sided alternative
• Ha m ¹ m0

102
Hypothesis testing
103
P-value

• The P-value provides the probability that the
  test statistic would take a value as extreme or
  more extreme than the value observed if the null
  hypothesis were true.
• The smaller the P-value, the stronger the
  evidence the data provide against the null
  hypothesis

104
P-value

• A measure of strength of evidence in support of
  the null hypothesis
• Large p-values indicate that the Ho is quite
  plausible given the data
• Small p-values indicate Ho is implausible, data
  are inconsistent with Ho
• Rejecting the null hypothesis usually implies a
  treatment effect or a real difference exists
• Strength of evidence is a continuous spectrum but
  we tend to use plt0.05 as the point at which we
  reject Ho
• The value of p which just rejects Ho is called a,
  the Type I error we will inappropriately reject
  Ho with risk a
• Always try to compute an exact p-value is
  possible (as opposed to saying p lt 0.05 or
  0.025ltplt0.05)

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Strength of evidence
106
Statistical Significance

• If the P-value is as small or smaller than the
  significance level a (i.e., P-value a), then we
  say that data are statistically significant at
  level a.
• If we choose a 0.05, we are requiring that the
  data give evidence against H0 so strong that it
  would occur no more than 5 of the time when H0
  is true.
• If we choose a 0.01, we are insisting on
  stronger evidence against H0, evidence so strong
  that it would occur only 1 of the time when H0
  is true.

107
One vs. two-sided tests

• Determined by the alternative hypothesis
• Two-sided test implies that we are interested in
  detecting departures from Ho in BOTH directions
• One-sided test implies that we are specifying a
  DIRECTIONAL alternative hypothesis (usually
  dictated by our understanding of the biology)
• Must specify direction of effect a-priori

108
One vs. two-sided tests

• Some people refuse to accept that one-sided
  tests are legitimate (e.g. NEJM, FDA)
• Statistical trickery
• Always possible that treatment has negative
  effect
• But
• Perfectly acceptable statistically speaking
• Accepting Ho does not rule out a negative
  treatment effect
• It is legitimate and often desirable not to
  prove that treatment is harmful

109
Inference about a population mean
110
Conditions for Inferenceabout a Mean

• Data are a SRS of size n
• Population has a Normal distribution with mean µ
  and standard s (both µ and s are unknown)
• Because s is unknown (realistic situation), we
  can NOT use z procedures for our confidence
  intervals and significance tests

111
Standard Error
When we do not know population standard deviation
s, we use sample standard deviation s to
calculate the standard error
This is called the standard error of the mean
112
One-Sample t Statistic

• The one-sample z statistic now becomes a
  one-sample t statistic
• The t statistic does NOT follow a Normal
  distribution
• It follows a t distribution with n 1 degrees
  of freedom

113
The t Distributions

• t distributions are a family of distributions
  similar to the Standard Normal Z distribution
• Each member of t identified by its degree of
  freedom
• Notation t(k) denotes a t distribution with k
  degrees of freedom

114
The t Distributions
As k increases, the t(k) curve approaches the Z
curve As n increases, s becomes a better
estimate of s
115
t Table
Table C gives t critical values with upper tail
probability p and corresponding confidence level C
The bottom row of the table applies to z because
a t with infinite degrees of freedom is a
standard Normal (Z) variable
116
One-Sample t Confidence Interval

• Take a SRS of size n from a population with
  unknown mean m and unknown standard deviation s.
  A level C confidence interval for m is given by

where t is the critical value with (n 1) for
confidence level C (from t Table)
117
One-Sample t Test

• The t test is similar in form to the z test
  learned earlier. The test statistic is

The test statistic has n 1 degrees of
freedom. Get the approximate P-value from the t
table.
118
Matched Pairs t Procedures

• Matched pair samples allow us to compare
  responses to two treatments in paired couples
• Apply the one-sample t procedures to the observed
  differences within pairs
• The parameter m is the mean difference in the
  responses to the two treatments within matched
  pairs in the entire population

119
Case Study Matched Pairs
Air Pollution

• Pollution index measurements were recorded for
  two areas of a city on each of 8 days
• To analyze, subtract Area B levels from Area A
  levels. The 8 differences form a single sample.
• Are the average pollution levels the same for the
  two areas of the city?

120
Normality Assumptiont Procedure Robustness

• t procedures produce perfect results when the
  population is Normal. They are robust, producing
  almost perfect confidence intervals and P-values
  when the sample is moderate to small
• Sample size less than 15 Use t procedures if
  the data appear about Normal (symmetric, single
  peak, no outliers). If the data are skewed or if
  outliers are present, do not use t.
• Sample size at least 15 The t procedures can be
  used except in the presence of outliers or strong
  skewness in the data.
• Large samples The t procedures can be used even
  for clearly skewed distributions when the sample
  is large, roughly n 40.

121
Can we use a t procedure?
Moderately sized data set (n 20), with strong
negative skew. t procedures cannot be trusted
122
Can we use t?

• This histogram shows the distribution of word
  lengths in Shakespeares plays. The sample is
  very large.
• The data has a strong positive skew, but there
  are no outliers. We can use the t procedures
  since n 40

123
Can we use t?
The distribution has no clear violations of
Normality. Therefore, we trust the t procedure.
124
2-sample problems
125
Conditions for inference comparing two means

• We have two independent SRSs (simple random
  samples) coming from two distinct populations
  (like men vs. women) with (m1,s1) and (m2,s2)
  unknown.
• Both populations should be Normally distributed.
  However, in practice, it is enough that the two
  distributions have similar shapes and that the
  sample data contain no strong outliers.

126
Two-sample t-test

• The null hypothesis is that both population means
  m1 and m2 are equal, thus their difference is
  equal to zero.
• H0 m1 m2 ltgt m1 - m2 0
• with either a one-sided or a two-sided
  alternative hypothesis.
• We find how many standard errors (SE) away from
  (m1 - m2) is ( 1 - 2) by standardizing with
  t
• Because in a two-sample test H0 poses (m1 - m2)
  0, we simply use
• with df smallest(n1 - 1,n2 - 1)

127
Two sample t-confidence interval

• Because we have two independent samples we use
  the difference between both sample averages (
  1 - 2) to estimate (m1 - m2).

• Practical use of t t
• C is the area between -t and t.
• We find t in the line of Table C for df
  smallest (n1-1 n2-1) and the column for
  confidence level C.
• The margin of error m is

128
Robustness

• The two-sample statistic is the most robust when
  both sample sizes are equal and both sample
  distributions are similar. But even when we
  deviate from this, two-sample tests tend to
  remain quite robust.
• As a guideline, a combined sample size (n1 n2)
  of 40 or more will allow you to work even with
  the most skewed distributions.

129
Two-sample test assuming equal variance

• There are two versions of the two-sample t-test
  one assuming equal variance (pooled two-sample
  test) and one not assuming equal variance
  (unequal variance) for the two populations. You
  may have noticed slightly different formulas and
  degrees of freedom.

The pooled (equal variance) two-sample t-test was
often used before computers because it has
exactly the t distribution for degrees of freedom
n1 n2 - 2. However, the assumption of equal
variance is hard to check, and thus the unequal
variance test is safer.
Two normally distributed populations with unequal
variances
130
Comparing two standard deviations

• It is also possible to compare two population
  standard deviations s1 and s2 by comparing the
  standard deviations of two SRSs. However, the
  procedures are not robust at all against
  deviations from normality.
• When s12 and s22 are sample variances from
  independent SRSs of sizes n1 and n2 drawn from
  normal populations, the F-statistic F s12 / s22
• has the F distribution with n1 - 1 and n2 - 1
  degrees of freedom when H0 s1 s2 is true.
• The F-value is then compared with critical values
  from Table D for the P-value with a one-sided
  alternative this P-value is doubled for a
  two-sided alternative.

131
Proportions
132
Two-Way Tables

• Cross-tabulate counts to form two-way table
• row variable
• column variable
• The count of observations falling into each
  combination of categories fall in tables cell
• Counts are totaled to create marginal totals

133
2 independent samples

• Described as a 2 x 2 table
• Exact test
• Fishers Exact test
• Approximate tests
• Z-test
• Chi-squared test
• Summary measures
• RD (risk difference) Ho RD 0
• RR Ho RR 1
• OR Ho OR 1

134
Fishers test or Chi-square?

• Either can be used for analyzing contingency
  tables with 2 rows and 2 colums
• Fishers test is the best choice as it always
  gives the exact P value
• Chi-square test is simpler to calculate but
  yields only an approximate P value
• If using computer, choose Fishers test
• Avoid chi-square when the numbers in the
  contingency table are very small (lt 6)

135
2 related (paired) samples

• Exact test
• Based on binomial distribution
• Approximate test
• McNemars Chi-squared test
• Summary measure
• OR estimator
• OR confidence interval

136
2 independent stratified samples

• 2 x 2 x k table
• Estimation of the OR
• Mantel-Haenszel chi-square (good because can
  handle cells with 0)
• Woolf (precision-weighted) chi-square (most
  commonly used for meta-analysis because formulas
  simple, but not as good as M-H))
• Peto (O-E-V) chi-square
• Generalized M-H and P-W estimators
• Tests for homogeneity over strata i.e. can ORs
  be pooled across strata?
• Exact test
• Zelens test
• Approximate chi-squared tests
• Breslow-Day
• Woolf/Precision-weighted

137
Recommended analysis of 2 x 2 x k tables

• Choose a suitable effect measure (RD, RR or OR)
• Test for homogeneity of stratum-specific effect
  measures
• Compute the summary effect measure estimate and
  its associated CI
• Test for association

138
Sample size calculation
139
http//statpages.org/

• (for all your statistical needs, including sample
  size calculations)

140
Nonparametric tests
141
Non-parametric tests

• Make no distributional assumptions about data
  (non-normal data)
• Main focus is p-value, dont lend themselves to
  CIs or sample size calculations
• Sign test equivalent to one-sample test
• Wilcoxon Rank Sum (Mann-Whitney Test)
  equivalent to independent two sample t-test
• Wilcoxon Signed Rank equivalent to the paired
  t-test

142
Non-parametric tests - advantages

• Fewer assumptions are required (i.e. no
  distributional assumptions or assumptions about
  equality of variances)
• Only nominal (categorical) or ordinal (ranked)
  data are required, rather than numerical
  (interval) data

143
Non-parametric tests - disadvantages

• They are less efficient
• Less powerful than parametric counterparts
• Often lead to overestimation of variances of test
  statistics when there are large proportions of
  tied observations
• They dont lend themselves easily to CIs and
  sample size calculations
• Interpretation of non-parametric results is quite
  hard

144
Scatterplots and correlation
145
Variable X and variable Y

• Relationship between 2 quantitative variables
• Explanatory variable X
• Response variable Y
• Does X cause Y ?

146
Scatterplot

• Start by plotting bivariate data points to make a
  scatterplot

147
Example of a scatterplot
X students taking SAT Y mean SAT verbal
score What is the relation between X and Y?
148
Interpreting scatterplots

• Form can data be described by a straight line?
• Exceptions (outliers) to form
• Direction upward or downward
• Strength extent to which data points adhere to
  trend line

149
Examples of Forms
150
Strength and direction

• Direction positive, negative, neither
• Strength How closely do points adhere to trend
  line?
• Close fitting ? strong
• Loose fitting ? weak

151
Strength cannot be judged by eye alone

• These two scatterplots are of the same data set
• The second scatter plot looks like a stronger
  correlation, but this is an artifact of the axis
  scaling

152
Correlation coefficient (r)

• r the correlation coefficient
• r is always between -1 and 1, inclusive
• r 1? all points on upward sloping line
• r -1 ? all points on downward line
• r 0 ? no line or horizontal line
• The closer r is to 1 or 1, the better the fit,
  the stronger the correlation

153
Interpretation of r
Direction positive or negative Strength the
closer r is to 1, the stronger the
correlation 0.0 ? r lt 0.3 ? weak
correlation 0.3 ? r lt 0.7 ? moderate
correlation 0.7 ? r lt 1.0 ? strong correlation
r 1.0 ? perfect correlation
154
Not all Relationships are Linear Miles per
Gallon versus Speed

• r ? 0 (flat line) with strong non-linear relation
• But

155
Not all Relationships are Linear Miles per
Gallon versus Speed

• Very strong non-linear (curved) relationship
• r was misleading!

156
Outliers and Correlation
The outlier in the above graph decreases r so
that r 0 If we remove the outlier ? strong
relation
157
Beware!

• Not all relations are linear
• Outliers can have large influence on r
• Lurking variables confound relations

158
Regression
159
Objectives of Regression

• Quantitative X (explanatory)
• Quantitative Y (response)
• Objectives
• Describe the change in Y per unit X
• Predict average Y at given X

160
Equation for an algebraic line
Y (intercept) (slope)(X) or Y (slope)(X)
(intercept)
Intercept ? where line crosses Y axisSlope ?
angle of line
161
Equation for a regression line

• Algebraic line ? every point falls on lineExact
  Y intercept (slope)(X)
• Statistical line ? scatter around linear trend
• Predicted Y intercept (slope)(X)

y a bx where y (y-hat) is the predicted
value of Y, a is the intercept, and b is the slope
162
What Line Fits Best?

• The method we use to draw the best fitting line
  is called the least squares method
• If we try to draw the line by eye, different
  people will draw different lines

163
The least squares regression line

• Each point has
• Residual observed y predicted y
• distance of point from line predicted by
  model

The least squares line minimizes the sum of the
square residuals
164
Regression Line

• For the bird population data
• a 31.9343
• b ?0.3040
• The linear regression equation is
• y 31.9343 ? 0.3040x

The slope -0.3040 represents the average change
in Y per unit X
165
CautionsAbout Correlation and Regression

• Describe only linear relationships
• Beware of influential outliers
• Cannot predict beyond the range of X (do not
  extrapolate)
• Beware of lurking variables (variables other than
  X and Y)
• Association does not equal causation!

166
Transformations

• Often we work with transformed data rather than
  the original numbers
• 1/x for skewed left data
• Log (x) or ln (x) when skewed right
• Transforming data can
• Make a skewed distribution more symmetric
• Make the distribution more normal-like
• Stabilize variability
• Linearize a relationship between 2 or more
  variables
• Show summary statistics in original units but
  test on the transformed scale

167
Caution

• Even strong correlations may be non-causal
  (Beware lurking variables!)

168
Survival analysis
169
One sample case

• Time-to-event data
• Estimating the survival curve Kaplan-Meier
  (product limit) method
• Inferences based on survival curve

170
Two-sample case

• Comparison at a fixed time
• Hazard rates
• Overall comparison
• Mantel-Haenszel method
• Logrank method
• Estimation of common hazard ratio
• Summary event rates
• Sample size

171
What statistical test should I use?
172
Inferential statistics
173
Inferential statistics
174
Multiple comparisons

• ANOVA, Kruskall-Wallis and chi-square will test
  whether there are any differences between groups
• If doing multiple comparisons, must correct
• Bonferroni ( p by number of comparisons)
• Student-Neuman-Keuls
• Tukeys
• Scheffes f test

175
Common errors in statistical interpretation
176
Error 1

• The p-value is the probability of the null
  hypothesis being true base on the observed result

177
Error 2

• Failure to reject the null hypothesis equals
  acceptance of the null hypothesis

178
Error 3

• Failure to reject the null hypothesis equals
  rejection of the alternate hypothesis

179
Error 4

• An a level of 0.05 is a standard with an
  objective basis

180
Error 5

• The smaller the p-value, the larger the effect

181
Error 6

• Statistical significance implies importance

182
Error 7

• Data verify or refute theory