@let@token what we learned (and how it's all...

What we learned(and how it’s all connected)

Eli Gurarie

Stat 311 - Lecture 25University of Washington - Seattle

March 15, 2013

From Lecture 1: Broad outline of course

1 Descriptive statistics:

visualizations and statistical summaries and experimental Design.

2 Probability theory

random processes and distributions and Central Limit Theorem.

3 Inference statistics:

Parameter estimates, hypothesis testing, and modeling.

From Lecture 1: Basic definitions

The research question.

A population is the collection of all possible individuals of interest.

A sample is a subset of the population.

An individual (or experimental unit) is an object on which observationsare made.

A variable is any quantity that can be measured on an individual.

A dependent or response variable the focus of our researchquestion, or very closely linked to itAn independent variable or covariate or explanatory factor is anadditional factor we consider for explaining the response

A dependent or response variable the focus of our researchquestion, or very closely linked to it

An independent variable or covariate or explanatory factor is anadditional factor we consider for explaining the response

Descriptive statistics

Visualization

Stem and leaf plots (quicksummary of small datasets)

Histograms (distribution of largenumbers)

Box-plots (relationship tocategorical covariates)

Scatter plots (relationship tocontinuous covariates)

Multi-dimensional plots

Numerical summaries: Statistics

Measures of center (sample mean,median, mode)

Measures of spread (samplestandard deviation, variance)

Measures of weirdness (skewness,outliers)

Describing relationships betweencontinuous covariates variables:

Correlations coefficientCoefficient of determinationSlope / Intercept estimates

Sums of squares:

of Model/Group(SSM/G )of Error/Residual(SSE/R)of Total variation (SST )

Visualization

Sums of squares:

Visualization

Sums of squares:

Visualization

Sums of squares:

Visualization

Sums of squares:

Visualization

Sums of squares:

Visualization

Sums of squares:

Visualization

Sums of squares:

Visualization

Sums of squares:

Visualization

Sums of squares:

Visualization

Sums of squares:

Experimental Design

Causality vs. Correlation

Precision vs. Accuracy (Bias, sample size)

Observational vs. Experimental studies

Randomization

Treatments

Control groups

Stratification / Blocking

Blinding

Balanced vs. Unbalanced design

Placebo

Replication

Confounding variables

Experimental Design

Randomization

Treatments

Control groups

Blinding

Placebo

Replication

Experimental Design

Randomization

Treatments

Control groups

Blinding

Placebo

Replication

Experimental Design

Randomization

Treatments

Control groups

Blinding

Placebo

Replication

Experimental Design

Randomization

Treatments

Control groups

Blinding

Placebo

Replication

Experimental Design

Randomization

Treatments

Control groups

Blinding

Placebo

Replication

Experimental Design

Randomization

Treatments

Control groups

Blinding

Placebo

Replication

Experimental Design

Randomization

Treatments

Control groups

Blinding

Placebo

Replication

Experimental Design

Randomization

Treatments

Control groups

Blinding

Placebo

Replication

Experimental Design

Randomization

Treatments

Control groups

Blinding

Placebo

Replication

Experimental Design

Randomization

Treatments

Control groups

Blinding

Placebo

Replication

Experimental Design

Randomization

Treatments

Control groups

Blinding

Placebo

Replication

Experimental Design

Randomization

Treatments

Control groups

Blinding

Placebo

Replication

From Lecture 9: Experiments can be very powerful!

Experiments

Are your best chance to test causal relationships

Good experimental design can be very tricky.

Controlling for all confounding or lurkingvariables is hard.But there are some design choices andtechniques (randomization, stratification,blocking, balance, blinding) that help thedata you collect answer the questions you areasking

If the design is good - then the analysis is trivial!

Lecture 10: Observational Studies Can Be TrickyEmage 1: 12/25/1996

estimate: 453K, CI: 181 − 3715K

Density (n/km^2)

Ice Concentration

6080100

50 100 150 200 250

Distance to Ice Edge (km)

0 50 100 150

0 20 40 60 80 120

10 20 30 40 50 60

0 50 100 150

0 50 100 200 300

50 100 150 200 250

lm(Dt ∼ OnShelf * (IC.r + IC + IC2) * (DIce.r + DIce + DIce2) - 1)

height Df Sum Sq Mean Sq F value Pr(>F)

OnShelf 2 195.80 97.90 4898.23 0.0000 ***IC.r 1 0.42 0.42 21.18 0.0000 ***IC 1 0.05 0.05 2.31 0.1291IC2 1 0.03 0.03 1.33 0.2488DIce.r 1 0.01 0.01 0.52 0.4722DIce 1 0.05 0.05 2.51 0.1140DIce2 1 1.52 1.52 75.97 0.0000 ***OnShelf:IC.r 1 0.02 0.02 1.24 0.2672OnShelf:IC 1 0.00 0.00 0.00 0.9592OnShelf:IC2 1 0.01 0.01 0.27 0.6024OnShelf:DIce.r 1 0.08 0.08 4.02 0.0458 *OnShelf:DIce 1 0.00 0.00 0.00 0.9907OnShelf:DIce2 1 0.01 0.01 0.57 0.4500IC.r:DIce.r 1 0.01 0.01 0.37 0.5409IC.r:DIce 1 0.01 0.01 0.52 0.4707IC.r:DIce2 1 0.00 0.00 0.17 0.6798IC:DIce.r 1 0.11 0.11 5.34 0.0214 *IC:DIce 1 0.00 0.00 0.24 0.6251IC:DIce2 1 0.01 0.01 0.33 0.5665IC2:DIce.r 1 0.09 0.09 4.69 0.0311 *IC2:DIce 1 0.00 0.00 0.03 0.8676IC2:DIce2 1 0.00 0.00 0.01 0.9202OnShelf:IC.r:DIce.r 1 0.05 0.05 2.74 0.0987 ·OnShelf:IC.r:DIce 1 0.05 0.05 2.34 0.1271OnShelf:IC.r:DIce2 1 0.18 0.18 8.79 0.0032 **OnShelf:IC:DIce.r 1 0.02 0.02 0.92 0.3391OnShelf:IC:DIce 1 0.00 0.00 0.05 0.8193OnShelf:IC:DIce2 1 0.00 0.00 0.12 0.7286OnShelf:IC2:DIce.r 1 0.07 0.07 3.46 0.0635 ·OnShelf:IC2:DIce 1 0.00 0.00 0.07 0.7884OnShelf:IC2:DIce2 1 0.04 0.04 2.02 0.1559Residuals 343 6.86 0.02

Probability Theory

Random variables

Parameters

Discrete distributions:

probability mass functions (pdf)Bernoulli, Binomial, relationships between them

Continuous distributions

Probability density functions (pdf)Normal/Gaussian, Uniform, T-distribution, F-distribution

Expectations / Variances

Central limit theorem

Probability Theory

Random variables

Parameters

Probability Theory

Random variables

Parameters

Probability Theory

Random variables

Parameters

Probability Theory

Random variables

Parameters

Probability Theory

Random variables

Parameters

Inference: General principles

Relationship between population parameters and sample statistics.

Sampling distributions of means and proportion - (variance known)

Confidence intervals / margin of error / point estimate

Significance level vs. PowerType I and Type II errorα, β

10 / 1

Sampling distribution - under H0 and HA

X ∼ N(µA,σA√n)

0 1 2 3 4 5 6 7

n = 4 X = 2.82

0 1 2 3 4 5 6 7

n = 8 X = 2.58

0 1 2 3 4 5 6 7

n = 16 X = 2.41

β = 12% β = 2.2% β = 0.07%

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Back in Lecture 20: What we’ve done so far...

We started with the coinflip

... derived the binomialdistribution

... obtained the CentralLimit Theorem

... calculated the sampling distribution ofthe mean: X ∼ N(µ, σ√

n) ... and developed the machinery of

hypothesis tests.

This is a big piece of classical statistics... and the best part is we derived(almost) all of the math behind it from “scratch”!

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Inference: Specifics

Sampling distribution of mean or proportion

σ known, large sample size, proportion

Z-test of means and difference of means and proportions

T-statistic (σ unknown)

T-tests of means and difference of means

Equal/unequal standard deviationEqual/unequal sample sizes

F-tests for ratios of variance estimates

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p-values!

http://xkcd.com/882/

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Comparing relationship between covariate X and Y

Models corresponding to different hypotheses of relationships.

Discrete groups X : One-way ANOVA

Continuous X : linear regression

Decomposition of sum of squares

Mean squares as estimates of variance

F-test of variance estimates

Degrees of freedom

Multiple covariates (2 or more way ANOVA)

Model selection

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Degrees of freedom

Model selection

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Degrees of freedom

Model selection

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Degrees of freedom

Model selection

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Degrees of freedom

Model selection

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Degrees of freedom

Model selection

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Degrees of freedom

Model selection

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Degrees of freedom

Model selection

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Degrees of freedom

Model selection

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Degrees of freedom

Model selection

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Statistics ...

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Statistics ...

Good luck!

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