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Answering Questions with Data
Introductory Statistics for Psychology Students
Matthew J. C. Crump, Danielle J. Navarro, Jeffrey Suzuki
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Front Matter
Abstract
Preface
1
Why Statistics?
1.1
On the psychology of statistics
1.1.1
The curse of belief bias
1.2
The cautionary tale of Simpson’s paradox
1.3
Statistics in psychology
1.4
Statistics in everyday life
1.5
There’s more to research methods than statistics
1.6
A brief introduction to research design
1.7
Introduction to psychological measurement
1.7.1
Some thoughts about psychological measurement
1.7.2
Operationalization: defining your measurement
1.8
Scales of measurement
1.8.1
Nominal scale
1.8.2
Ordinal scale
1.8.3
Interval scale
1.8.4
Ratio scale
1.8.5
Continuous versus discrete variables
1.8.6
Some complexities
1.9
Assessing the reliability of a measurement
1.10
The role of variables: predictors and outcomes
1.11
Experimental and non-experimental research
1.11.1
Experimental research
1.11.2
Non-experimental research
1.12
Assessing the validity of a study
1.12.1
Internal validity
1.12.2
External validity
1.12.3
Construct validity
1.12.4
Face validity
1.12.5
Ecological validity
1.13
Confounds, artifacts and other threats to validity
1.13.1
History effects
1.13.2
Maturation effects
1.13.3
Repeated testing effects
1.13.4
Selection bias
1.13.5
Differential attrition
1.13.6
Non-response bias
1.13.7
Regression to the mean
1.13.8
Experimenter bias
1.13.9
Demand effects and reactivity
1.13.10
Placebo effects
1.13.11
Situation, measurement and subpopulation effects
1.13.12
Fraud, deception and self-deception
1.14
Summary
1.15
Videos
1.15.1
Terms of Statistics
2
Describing Data
2.1
This is what too many numbers looks like
2.2
Look at the data
2.2.1
Stop, plotting time (o o oh) U can plot this
2.2.2
Histograms
2.3
Important Ideas: Distribution, Central Tendency, and Variance
2.4
Measures of Central Tendency (Sameness)
2.4.1
From many numbers to one
2.4.2
Mode
2.4.3
Median
2.4.4
Mean
2.4.5
What does the mean mean?
2.4.6
All together now
2.5
Measures of Variation (Different
ness
)
2.5.1
The Range
2.5.2
The Difference Scores
2.5.3
The Variance
2.5.3.1
Deviations from the mean, Difference scores from the mean
2.5.3.2
The mean is the balancing point in the data
2.5.3.3
The squared deviations
2.5.3.4
Finally, the variance
2.5.4
The Standard Deviation
2.6
Using Descriptive Statistics with data
2.7
Rolling your own descriptive statistics
2.7.1
Absolute deviations
2.7.2
Other sign-inverting operations
2.8
Remember to look at your data
2.8.1
Anscombe’s Quartet
2.8.2
Datasaurus Dozen
2.9
Videos
2.9.1
Measures of center: Mode
2.9.2
Measures of center: Median and Mean
2.9.3
Standard deviation part I
2.9.4
Standard deviation part II
3
Correlation
3.1
If something caused something else to change, what would that look like?
3.1.1
Charlie and the Chocolate factory
3.1.2
Scatter plots
3.1.3
Positive, Negative, and No-Correlation
3.2
Pearson’s r
3.2.1
The idea of co-variance
3.3
Turning the numbers into a measure of co-variance
3.3.1
Co-variance, the measure
3.3.2
Pearson’s r we there yet
3.4
Examples with Data
3.5
Regression: A mini intro
3.5.1
The best fit line
3.5.2
Lines
3.5.3
Computing the best fit line
3.6
Interpreting Correlations
3.6.1
Correlation does not equal causation
3.6.1.1
Even when there is causation, there might not be obvious correlation
3.6.1.2
Confounding variable, or Third variable problem
3.6.2
Correlation and Random chance
3.6.2.1
Monte-carlo simulation of random correlations
3.6.2.2
Increasing sample-size decreases opportunity for spurious correlation
3.6.3
Some more movies
3.6.3.1
Watching how correlation behaves when there is no correlation
3.6.3.2
Watching correlations behave when there really is a correlation
3.7
Summary
4
Probability, Sampling, and Estimation
4.1
How are probability and statistics different?
4.2
What does probability mean?
4.2.1
The frequentist view
4.2.2
The Bayesian view
4.2.3
What’s the difference? And who is right?
4.3
Basic probability theory
4.3.1
Introducing probability distributions
4.4
The binomial distribution
4.4.1
Introducing the binomial
4.4.2
Working with the binomial distribution in R
4.5
The normal distribution
4.5.1
Probability density
4.6
Other useful distributions
4.7
Summary of Probability
4.8
Samples, populations and sampling
4.8.1
Defining a population
4.8.2
Simple random samples
4.8.3
Most samples are not simple random samples
4.8.4
How much does it matter if you don’t have a simple random sample?
4.8.5
Population parameters and sample statistics
4.9
The law of large numbers
4.10
Sampling distributions and the central limit theorem
4.10.1
Sampling distribution of the sample means
4.10.2
Seeing the pieces
4.10.3
Sampling distributions exist for any sample statistic!
4.11
The central limit theorem
4.12
z-scores
4.12.1
Idea behind z-scores
4.12.2
Calculating z-scores
4.13
Estimating population parameters
4.13.1
Concrete population parameters
4.13.2
Abstract population parameters
4.13.2.1
Complications with inference
4.13.3
Experiments and Population parameters
4.13.4
Interim summary
4.13.5
Estimating the population mean
4.13.6
Estimating the population standard deviation
4.14
Estimating a confidence interval
4.14.1
A slight mistake in the formula
4.15
Summary
4.16
Videos
4.16.1
Introduction to Probability
4.16.2
Chebychev’s Theorem
4.16.3
Z-scores
4.16.4
Normal Distribution I
4.16.5
Normal Distribution II
5
Foundations for inference
5.1
Brief review of Experiments
5.2
The data came from a distribution
5.2.1
Uniform distribution
5.2.2
Not all samples are the same, they are usually quite different
5.2.3
Large samples are more like the distribution they came from
5.3
Is there a difference?
5.3.1
Chance can produce differences
5.3.2
Differences due to chance can be simulated
5.4
Chance makes some differences more likely than others
5.5
The Crump Test
5.5.1
Intuitive methods
5.5.2
Part 1: Frequency based intuition about occurrence
5.5.3
Part 2: Simulating chance
5.5.4
Part 3: Judgment and Decision-making
5.5.4.1
Grey areas
5.5.4.2
Making decisions and being wrong
5.5.5
Part 4: Experiment Design
5.5.6
Part 5: I have the power
5.5.7
Summary of Crump Test
5.6
The randomization test (permutation test)
5.6.1
Pretend example does chewing gum improve your grades?
5.6.1.1
Doing the randomization
5.6.1.2
Simulating the mean differences across the different randomizations
5.6.2
Take homes so far
5.7
Videos
5.7.1
Null and Alternate Hypotheses
5.7.2
Types of Errors
6
t-tests
6.1
Check your confidence in your mean
6.2
One-sample t-test: A new t-test
6.2.1
Formulas for one-sample t-test
6.2.2
What does t represent?
6.2.3
Calculating t from data
6.2.4
How does t behave?
6.2.5
Making a decision
6.3
Paired-samples t-test
6.3.1
Mehr, Song, and Spelke (2016)
6.3.2
The data
6.3.3
The difference scores
6.3.4
The mean difference
6.3.5
Calculate t
6.3.6
Interpreting
\(t\)
s
6.3.7
Getting the p-values for
\(t\)
-values
6.3.8
One-tailed tests
6.3.9
Two-tailed tests
6.3.10
One or two tailed, which one?
6.3.11
Degrees of freedom
6.3.1
The paired samples t-test strikes back
6.4
Independent samples t-test: The return of the t-test?
6.5
Simulating data for t-tests
6.5.1
Simulating a one-sample t-test
6.5.2
Simulating a paired samples t-test
6.5.3
Simulating an independent samples t.test
6.6
Videos
6.6.1
One or Two tailed tests
7
ANOVA
7.1
ANOVA is Analysis of Variance
7.2
One-factor ANOVA
7.2.1
Computing the
\(F\)
-value
7.2.2
SS Total
7.2.3
SS Effect
7.2.4
SS Error
7.2.5
Degrees of freedom
7.2.6
Mean Squared Error
7.2.7
Calculate F
7.2.8
The ANOVA TABLE
7.3
What does F mean?
7.3.1
Making Decisions
7.3.2
Fs and means
7.3.2.1
ANOVA is an omnibus test
7.3.2.2
Looking at a bunch of group means
7.3.2.3
Looking at bar graphs
7.3.2.4
What mean differences look like when
\(F\)
is less than 1
7.3.2.5
What mean differences look like when F > 3.35
7.4
ANOVA on Real Data
7.4.1
Tetris and bad memories
7.4.2
Comparing means after the ANOVA
7.4.2.1
Control vs. Reactivation+Tetris
7.4.2.2
Control vs. Tetris_only
7.5
ANOVA Summary
8
Repeated Measures ANOVA
8.1
Repeated measures design
8.2
Partitioning the Sums of Squares
8.3
Calculating the RM ANOVA
8.3.1
SS Total
8.3.2
SS Effect
8.3.3
SS Error (within-conditions)
8.3.4
SS Subjects
8.3.5
SS Error (left-over)
8.3.6
Check our work
8.3.7
Compute the MSEs
8.3.8
Compute F
8.3.9
p-value
8.4
Things worth knowing
8.4.1
Repeated vs between-subjects ANOVA
8.4.2
repeated measures designs are more sensitive
8.5
Real Data
8.6
Summary
9
Factorial ANOVA
9.1
Factorial basics
9.1.1
2x2 Designs
9.1.2
Factorial Notation
9.1.3
2 x 3 designs
9.2
Purpose of Factorial Designs
9.2.1
Factorials manipulate an effect of interest
9.2.2
Spot the difference
9.2.3
Distraction manipulation
9.2.4
Distraction effect
9.2.5
Manipulating the Distraction effect
9.3
Graphing the means
9.4
Knowing what you want to find out
9.5
Simple analysis of 2x2 repeated measures design
9.5.1
Main effects
9.5.2
Interaction
9.5.3
Looking at the data
9.5.4
Main effect of Distraction
9.5.5
Main effect of Reward
9.5.6
Interaction between Distraction and Reward
9.5.7
Writing it all up
9.5.8
2x2 Repeated Measures ANOVA
9.6
2x2 Between-subjects ANOVA
9.6.1
SS Total
9.6.2
SS Distraction
9.6.3
SS Reward
9.6.4
SS Distraction by Reward
9.6.5
SS Error
9.6.6
Check your work
9.7
Fireside chat
9.8
Real Data
9.8.1
Stand at attention
9.8.2
Plot the data
9.8.3
Conduct the ANOVA
9.8.4
Main effect of Congruency
9.8.5
Main effect of Posture
9.8.6
Congruency X Posture Interaction
9.8.7
What does it all mean?
9.9
Factorial summary
10
More On Factorial Designs
10.1
Looking at main effects and interactions
10.1.1
2x2 designs
10.2
Interpreting main effects and interactions
10.2.1
A consistent main effect and an interaction
10.2.2
An inconsistent main effect and an interaction
10.3
Mixed Designs
10.4
More complicated designs
10.4.1
2x3 design
10.4.2
2x2x2 designs
11
Simulating Data
11.1
Reasons to simulate
11.2
Simulation overview
11.3
Simulating t-tests
11.4
Simulating one-factor ANOVAs
11.5
Other resources
12
Thinking about answering questions with data
12.1
Effect-size and power
12.1.1
Chance vs. real effects
12.1.2
Effect size: concrete vs. abstract notions
12.1.3
Cohen’s d
12.2
Power
12.2.1
A digression about hypothesis testing
12.2.2
Back to power
12.2.3
Power curves
12.3
Planning your design
12.4
Some considerations
12.4.1
Low powered studies
12.4.2
Large N and small effects
12.4.3
Small N and Large effects
12.4.4
Type I errors are convincing when N is small
13
GIFs
13.1
Correlation GIFs
13.1.1
N=10, both variables drawn from a uniform distribution
13.1.2
Correlation between random deviates from uniform distribution across four sample sizes
13.1.3
Correlation between random deviates from normal distribution across four sample sizes
13.1.4
Correlation between X and Y variables that have a true correlation as a function of sample-size
13.1.5
Type I errors, sampling random deviates from normal distribution with regression lines
13.1.6
Cell-size and correlation
13.1.7
Regression
13.2
Sampling distributions
13.2.1
Sampling from a uniform distribution
13.2.2
Sampling from uniform with line showing expected value for each number
13.2.3
Sampling distribution of the mean, Normal population distribution and sample histograms
13.2.4
Null and True effect samples and sampling means
13.3
Statistical Inference
13.3.1
Randomization Test
13.3.2
Independent t-test Null
13.3.3
Independent t-test True
13.3.4
T-test True sample-size
13.3.5
one-factor ANOVA Null
13.3.6
Factorial Null
13.4
Distributions
13.4.1
Normal changing mean
13.4.2
Normal changing sd
Back Matter
References
Index
Colophon
Section
4.6
Other useful distributions
There are many other useful distributions, these include the
t
distribution, the
F
distribution, and the chi squared distribution. We will soon discover more about the
t
and
F
distributions when we discuss t-tests and ANOVAs in later chapters.
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