Small-N designs & Basic Statistical Concepts

Small-N designs & Basic Statistical Concepts
Psych 231: Research Methods in Psychology

Small N designs What are they?
Historically, these were the typical kind of design used until 1920’s when there was a shift to using larger sample sizes Even today, in some sub-areas, using small N designs is common place (e.g., clinical settings, psychophysics, expertise, etc.) Small N designs

Small N designs One or a few participants
Data are typically not analyzed statistically; rather rely on visual interpretation of the data Observations begin in the absence of treatment (BASELINE) Then treatment is implemented and changes in frequency, magnitude, or intensity of behavior are recorded These typically are experiments (involve manipulation by the researcher) Small N designs

Small N designs Baseline experiments – the basic idea is to show:
when the IV occurs, you get the effect when the IV doesn’t occur, you don’t get the effect (reversibility) Before introducing treatment (IV), baseline needs to be stable Measure level and trend Small N designs

Small N designs Level – how frequent (how intense) is behavior?
Are all the data points high or low? Trend – does behavior seem to increase (or decrease) Are data points “flat” or on a slope? Small N designs

ABA design ABA design (baseline, treatment, baseline)
The reversibility is necessary, otherwise something else may have caused the effect other than the IV (e.g., history, maturation, etc.) ABA design

Small N designs Advantages
Focus on individual performance, not fooled by group averaging effects Focus is on big effects (small effects typically can’t be seen without using large groups) Avoid some ethical problems – e.g., with non-treatments Allows to look at unusual (and rare) types of subjects (e.g., case studies of amnesics, experts vs. novices) Often used to supplement large N studies, with more observations on fewer subjects Small N designs

Small N designs Disadvantages
Effects may be small relative to variability of situation so NEED more observation Some effects are by definition between subjects Treatment leads to a lasting change, so you don’t get reversals Difficult to determine how generalizable the effects are Small N designs

Some researchers have argued that Small N designs are the best way to go.
The goal of psychology is to describe behavior of an individual Looking at data collapsed over groups “looks” in the wrong place Need to look at the data at the level of the individual Small N designs

Statistics Mistrust of statistics? It is all in how you use them
They are a critical tool in research Statistics

Statistics Why do we use them? Descriptive statistics
Used to describe, simplify, & organize data sets Describing distributions of scores Inferential statistics Used to test claims about the population, based on data gathered from samples Takes sampling error into account, are the results above and beyond what you’d expect by random chance Statistics

Recall that a variable is a characteristic that can take different values.
The distribution of a variable is a summary of all the different values of a variable Both type (each value) and token (each instance) How much do you like psy231? Hate it Love it 5 values (1, 2, 3, 4, 5) 7 tokens (1,1,2,3,4,5,5) 1 5 5 4 1 3 2 Distribution

Distribution Many important distributions 5 1 2 5 3 Population Sample
All the scores of interest Sample All of the scores observed (your data) Used to estimate population characteristics Distribution of sample distributions Used to estimate sampling error Population 1 2 3 3 1 5 1 2 3 1 Sample How do we describe these distributions? Use descriptive statistics, focus on 3 properties Distribution

Distribution Properties of a distribution Shape Center
Symmetric v. asymmetric (skew) Unimodal v. multimodal Center Where most of the data in the distribution are Mean, Median, Mode Spread (variability) How similar/dissimilar are the scores in the distribution? Standard deviation (variance), Range Distribution

Visual descriptions - A picture of the distribution is usually helpful
Gives a good sense of the properties of the distribution Many different ways to display distribution Graphs Continuous variable: histogram, line graph (frequency polygons) Categorical variable: pie chart, bar chart Table Frequency distribution table Numerical descriptions of distributions Distribution

Graph for continuous variables
A frequency histogram Example: Distribution of scores on an exam Graph for continuous variables

Graph for continuous variables
A line graph Example: Distribution of scores on an exam Graph for continuous variables

Graphs for categorical variables
Bar chart Pie chart Graphs for categorical variables

Caution Be careful using a line graph for categorical variables
The line implies that there are responses between Smith and Doe, but there are not Caution

Frequency distribution table
Values (types) Counts Percentages Frequency distribution table

Properties of distributions: Shape
Symmetric The two sides line up Asymmetric (skewed) The two sides do not line up Properties of distributions: Shape

Symmetric Asymmetric (skewed) Negative Skew Positive Skew tail tail Properties of distributions: Shape

Unimodal (one mode) Major mode Minor mode Multimodal Bimodal examples Properties of distributions: Shape

Properties of distributions: Center
There are three main measures of center Mean (M): the arithmetic average Add up all of the scores and divide by the total number Most used measure of center Median (Mdn): the middle score in terms of location The score that cuts off the top 50% of the from the bottom 50% Good for skewed distributions (e.g. net worth) Mode: the most frequent score Good for nominal scales (e.g. eye color) A must for multi-modal distributions Properties of distributions: Center

The Mean The most commonly used measure of center
The arithmetic average Computing the mean Divide by the total number in the population The formula for the population mean is (a parameter): Add up all of the X’s The formula for the sample mean is (a statistic): Divide by the total number in the sample The Mean

Spread (Variability) How similar are the scores?
Range: the maximum value - minimum value Only takes two scores from the distribution into account Influenced by extreme values (outliers) Standard deviation (SD): (essentially) the average amount that the scores in the distribution deviate from the mean Takes all of the scores into account Also influenced by extreme values (but not as much as the range) Variance: standard deviation squared Spread (Variability)

Variability Low variability High variability
The scores are fairly similar High variability The scores are fairly dissimilar mean mean Variability

The standard deviation is the most popular and most important measure of variability.
The standard deviation measures how far off all of the individuals in the distribution are from a standard, where that standard is the mean of the distribution. Essentially, the average of the deviations. m Standard deviation

An Example: Computing the Mean
Our population 2, 4, 6, 8 m An Example: Computing the Mean

An Example: Computing Standard Deviation (population)
Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution. Our population 2, 4, 6, 8 m -3 X -  = deviation scores 2 - 5 = -3 An Example: Computing Standard Deviation (population)

Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution. Our population 2, 4, 6, 8 m -1 X -  = deviation scores 2 - 5 = -3 4 - 5 = -1 An Example: Computing Standard Deviation (population)

Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution. Our population 2, 4, 6, 8 m 1 X -  = deviation scores 2 - 5 = -3 6 - 5 = +1 4 - 5 = -1 An Example: Computing Standard Deviation (population)

Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution. Our population 2, 4, 6, 8 m 3 X -  = deviation scores 2 - 5 = -3 6 - 5 = +1 Notice that if you add up all of the deviations they must equal 0. 4 - 5 = -1 8 - 5 = +3 An Example: Computing Standard Deviation (population)

Step 2: So what we have to do is get rid of the negative signs. We do this by squaring the deviations and then taking the square root of the sum of the squared deviations (SS). SS =  (X - )2 2 - 5 = -3 4 - 5 = -1 6 - 5 = +1 8 - 5 = +3 X -  = deviation scores = (-3)2 + (-1)2 + (+1)2 + (+3)2 = = 20 An Example: Computing Standard Deviation (population)

Step 3: ComputeVariance (which is simply the average of the squared deviations (SS)) So to get the mean, we need to divide by the number of individuals in the population. variance = 2 = SS/N An Example: Computing Standard Deviation (population)

Step 4: Compute Standard Deviation To get this we need to take the square root of the population variance. standard deviation =  = An Example: Computing Standard Deviation (population)

To review: Step 1: Compute deviation scores Step 2: Compute the SS Step 3: Determine the variance Take the average of the squared deviations Divide the SS by the N Step 4: Determine the standard deviation Take the square root of the variance An Example: Computing Standard Deviation (population)

An Example: Computing Standard Deviation (sample)
To review: Step 1: Compute deviation scores Step 2: Compute the SS Step 3: Determine the variance Take the average of the squared deviations Divide the SS by the N-1 Step 4: Determine the standard deviation Take the square root of the variance This is done because samples are biased to be less variable than the population. This “correction factor” will increase the sample’s SD (making it a better estimate of the population’s SD) An Example: Computing Standard Deviation (sample)

Relationships between variables
Example: Suppose that you notice that the more you study for an exam, the better your score typically is. This suggests that there is a relationship between study time and test performance. We call this relationship a correlation. Relationships between variables

Relationships between variables
Properties of a correlation Form (linear or non-linear) Direction (positive or negative) Strength (none, weak, strong, perfect) To examine this relationship you should: Make a scatterplot Compute the Correlation Coefficient Relationships between variables

Scatterplot Plots one variable against the other
Useful for “seeing” the relationship Form, Direction, and Strength Each point corresponds to a different individual Imagine a line through the data points Scatterplot

Y X 1 2 3 4 5 6 Hours study X Exam perf. Y 6 1 2 5 3 4 Scatterplot

Correlation Coefficient
A numerical description of the relationship between two variables For relationship between two continuous variables we use Pearson’s r It basically tells us how much our two variables vary together As X goes up, what does Y typically do X, Y X, Y X, Y Correlation Coefficient

Linear Non-linear Form

Direction Positive Negative Y X Y X As X goes up, Y goes up
X & Y vary in the same direction Positive Pearson’s r As X goes up, Y goes down X & Y vary in opposite directions Negative Pearson’s r Direction

Strength Zero means “no relationship”.
The farther the r is from zero, the stronger the relationship The strength of the relationship Spread around the line (note the axis scales) Strength

r = 0.0 “no relationship” r = 1.0 “perfect positive corr.” r = -1.0 “perfect negative corr.” -1.0 0.0 +1.0 The farther from zero, the stronger the relationship Strength

Rel A Rel B r = -0.8 r = 0.5 -.8 .5 -1.0 0.0 +1.0 Which relationship is stronger? Rel A, -0.8 is stronger than +0.5 Strength

Compute the equation for the line that best fits the data points
Y X 1 2 3 4 5 6 Y = (X)(slope) + (intercept) 0.5 2.0 Change in Y Change in X = slope Regression

Regression Can make specific predictions about Y based on X X = 5
Y = (X)(.5) + (2.0) Y X 1 2 3 4 5 6 Y = (5)(.5) + (2.0) Y = = 4.5 4.5 Regression

Regression Also need a measure of error Y = X(.5) + (2.0) + error
Same line, but different relationships (strength difference) Y X 1 2 3 4 5 6 Y X 1 2 3 4 5 6 Regression

Cautions with correlation & regression
Don’t make causal claims Don’t extrapolate Extreme scores (outliers) can strongly influence the calculated relationship Cautions with correlation & regression

Inferential Statistics
Purpose: To make claims about populations based on data collected from samples What’s the big deal? Example Experiment: Group A - gets treatment to improve memory Group B - gets no treatment (control) After treatment period test both groups for memory Results: Group A’s average memory score is 80% Group B’s is 76% Is the 4% difference a “real” difference (statistically significant) or is it just sampling error? Inferential Statistics

Testing Hypotheses Step 1: State your hypotheses
Step 2: Set your decision criteria Step 3: Collect your data from your sample(s) Step 4: Compute your test statistics Step 5: Make a decision about your null hypothesis “Reject H0” “Fail to reject H0” Testing Hypotheses

This is the hypothesis that you are testing Null hypothesis (H0) Alternative hypothesis(ses) “There are no differences (effects)” Generally, “not all groups are equal” You aren’t out to prove the alternative hypothesis (although it feels like this is what you want to do) If you reject the null hypothesis, then you’re left with support for the alternative(s) (NOT proof!) Testing Hypotheses

In our memory example experiment Null H0: mean of Group A = mean of Group B Alternative HA: mean of Group A ≠ mean of Group B (Or more precisely: Group A > Group B) It seems like our theory is that the treatment should improve memory. That’s the alternative hypothesis. That’s NOT the one the we’ll test with inferential statistics. Instead, we test the H0 Testing Hypotheses

Step 2: Set your decision criteria Your alpha level will be your guide for when to: “reject the null hypothesis” “fail to reject the null hypothesis” This could be correct conclusion or the incorrect conclusion Two different ways to go wrong Type I error: saying that there is a difference when there really isn’t one (probability of making this error is “alpha level”) Type II error: saying that there is not a difference when there really is one Testing Hypotheses

Error types Real world (‘truth’) H0 is correct H0 is wrong
Type I error Reject H0 Experimenter’s conclusions Fail to Reject H0 Type II error Error types

Error types: Courtroom analogy
Real world (‘truth’) Defendant is innocent Defendant is guilty Type I error Find guilty Jury’s decision Type II error Find not guilty Error types: Courtroom analogy

Type I error: concluding that there is an effect (a difference between groups) when there really isn’t. Sometimes called “significance level” We try to minimize this (keep it low) Pick a low level of alpha Psychology: 0.05 and 0.01 most common Type II error: concluding that there isn’t an effect, when there really is. Related to the Statistical Power of a test How likely are you able to detect a difference if it is there Error types

Step 2: Set your decision criteria Step 3: Collect your data from your sample(s) Step 4: Compute your test statistics Descriptive statistics (means, standard deviations, etc.) Inferential statistics (t-tests, ANOVAs, etc.) Step 5: Make a decision about your null hypothesis Reject H0 “statistically significant differences” Fail to reject H0 “not statistically significant differences” Testing Hypotheses

Statistical significance
“Statistically significant differences” When you “reject your null hypothesis” Essentially this means that the observed difference is above what you’d expect by chance “Chance” is determined by estimating how much sampling error there is Factors affecting “chance” Sample size Population variability Statistical significance

(Pop mean - sample mean)
Population mean Population Distribution x Sampling error (Pop mean - sample mean) n = 1 Sampling error

Population mean Population Distribution Sample mean x x Sampling error (Pop mean - sample mean) n = 2 Sampling error

Generally, as the sample size increases, the sampling error decreases Population mean Population Distribution Sample mean x Sampling error (Pop mean - sample mean) n = 10 Sampling error

Typically the narrower the population distribution, the narrower the range of possible samples, and the smaller the “chance” Large population variability Small population variability Sampling error

Sampling error Population Distribution of sample means
These two factors combine to impact the distribution of sample means. The distribution of sample means is a distribution of all possible sample means of a particular sample size that can be drawn from the population Population Distribution of sample means XC Samples of size = n XA XD Avg. Sampling error XB “chance” Sampling error

Significance “A statistically significant difference” means:
the researcher is concluding that there is a difference above and beyond chance with the probability of making a type I error at 5% (assuming an alpha level = 0.05) Note “statistical significance” is not the same thing as theoretical significance. Only means that there is a statistical difference Doesn’t mean that it is an important difference Significance

Non-Significance Failing to reject the null hypothesis
Generally, not interested in “accepting the null hypothesis” (remember we can’t prove things only disprove them) Usually check to see if you made a Type II error (failed to detect a difference that is really there) Check the statistical power of your test Sample size is too small Effects that you’re looking for are really small Check your controls, maybe too much variability Non-Significance

From last time XA XB About populations Example Experiment:
Group A - gets treatment to improve memory Group B - gets no treatment (control) After treatment period test both groups for memory Results: Group A’s average memory score is 80% Group B’s is 76% H0: A = B H0: there is no difference between Grp A and Grp B Real world (‘truth’) H0 is correct H0 is wrong Experimenter’s conclusions Reject H0 Fail to Reject H0 Type I error Type II error Is the 4% difference a “real” difference (statistically significant) or is it just sampling error? Two sample distributions XA XB 76% 80% From last time

“Generic” statistical test
Tests the question: Are there differences between groups due to a treatment? H0 is true (no treatment effect) Real world (‘truth’) H0 is correct H0 is wrong Experimenter’s conclusions Reject H0 Fail to Reject H0 Type I error Type II error Two possibilities in the “real world” One population Two sample distributions XA XB 76% 80% “Generic” statistical test

Tests the question: Are there differences between groups due to a treatment? Real world (‘truth’) H0 is correct H0 is wrong Experimenter’s conclusions Reject H0 Fail to Reject H0 Type I error Type II error Two possibilities in the “real world” H0 is true (no treatment effect) H0 is false (is a treatment effect) Two populations XA XB XB XA 76% 80% 76% 80% People who get the treatment change, they form a new population (the “treatment population) “Generic” statistical test

XB XA ER: Random sampling error ID: Individual differences (if between subjects factor) TR: The effect of a treatment Why might the samples be different? (What is the source of the variability between groups)? “Generic” statistical test

XB XA ER: Random sampling error ID: Individual differences (if between subjects factor) TR: The effect of a treatment The generic test statistic - is a ratio of sources of variability Observed difference TR + ID + ER ID + ER Computed test statistic = = Difference from chance “Generic” statistical test

Sampling error Population “chance” Distribution of sample means
The distribution of sample means is a distribution of all possible sample means of a particular sample size that can be drawn from the population Population Distribution of sample means XC Samples of size = n XA XD Avg. Sampling error XB “chance” Sampling error

The generic test statistic distribution To reject the H0, you want a computed test statistics that is large reflecting a large Treatment Effect (TR) What’s large enough? The alpha level gives us the decision criterion TR + ID + ER ID + ER Distribution of the test statistic Test statistic Distribution of sample means -level determines where these boundaries go “Generic” statistical test

The generic test statistic distribution To reject the H0, you want a computed test statistics that is large reflecting a large Treatment Effect (TR) What’s large enough? The alpha level gives us the decision criterion Distribution of the test statistic Reject H0 Fail to reject H0 “Generic” statistical test

The generic test statistic distribution To reject the H0, you want a computed test statistics that is large reflecting a large Treatment Effect (TR) What’s large enough? The alpha level gives us the decision criterion Distribution of the test statistic Reject H0 “One tailed test”: sometimes you know to expect a particular difference (e.g., “improve memory performance”) Fail to reject H0 “Generic” statistical test

Things that affect the computed test statistic Size of the treatment effect The bigger the effect, the bigger the computed test statistic Difference expected by chance (sample error) Sample size Variability in the population “Generic” statistical test

Significance “A statistically significant difference” means:
the researcher is concluding that there is a difference above and beyond chance with the probability of making a type I error at 5% (assuming an alpha level = 0.05) Note “statistical significance” is not the same thing as theoretical significance. Only means that there is a statistical difference Doesn’t mean that it is an important difference Significance

Non-Significance Failing to reject the null hypothesis
Generally, not interested in “accepting the null hypothesis” (remember we can’t prove things only disprove them) Usually check to see if you made a Type II error (failed to detect a difference that is really there) Check the statistical power of your test Sample size is too small Effects that you’re looking for are really small Check your controls, maybe too much variability Non-Significance

Some inferential statistical tests
1 factor with two groups T-tests Between groups: 2-independent samples Within groups: Repeated measures samples (matched, related) 1 factor with more than two groups Analysis of Variance (ANOVA) (either between groups or repeated measures) Multi-factorial Factorial ANOVA Some inferential statistical tests

T-test Design Formula: Observed difference X1 - X2 T =
2 separate experimental conditions Degrees of freedom Based on the size of the sample and the kind of t-test Formula: Observed difference T = X X2 Diff by chance Based on sample error Computation differs for between and within t-tests T-test

T-test Reporting your results
The observed difference between conditions Kind of t-test Computed T-statistic Degrees of freedom for the test The “p-value” of the test “The mean of the treatment group was 12 points higher than the control group. An independent samples t-test yielded a significant difference, t(24) = 5.67, p < 0.05.” “The mean score of the post-test was 12 points higher than the pre-test. A repeated measures t-test demonstrated that this difference was significant significant, t(12) = 5.67, p < 0.05.” T-test

Analysis of Variance XB XA XC Designs Test statistic is an F-ratio
More than two groups 1 Factor ANOVA, Factorial ANOVA Both Within and Between Groups Factors Test statistic is an F-ratio Degrees of freedom Several to keep track of The number of them depends on the design Analysis of Variance

Analysis of Variance More than two groups F-ratio = XB XA XC
Now we can’t just compute a simple difference score since there are more than one difference So we use variance instead of simply the difference Variance is essentially an average difference Observed variance Variance from chance F-ratio = Analysis of Variance

1 factor ANOVA 1 Factor, with more than two levels XB XA XC
Now we can’t just compute a simple difference score since there are more than one difference A - B, B - C, & A - C 1 factor ANOVA

1 factor ANOVA The ANOVA tests this one!! XA = XB = XC XA ≠ XB ≠ XC
Null hypothesis: H0: all the groups are equal The ANOVA tests this one!! XA = XB = XC Do further tests to pick between these Alternative hypotheses HA: not all the groups are equal XA ≠ XB ≠ XC XA ≠ XB = XC XA = XB ≠ XC XA = XC ≠ XB 1 factor ANOVA

1 factor ANOVA Planned contrasts and post-hoc tests:
- Further tests used to rule out the different Alternative hypotheses XA ≠ XB ≠ XC Test 1: A ≠ B XA = XB ≠ XC Test 2: A ≠ C XA ≠ XB = XC Test 3: B = C XA = XC ≠ XB 1 factor ANOVA

1 factor ANOVA Reporting your results The observed differences
Kind of test Computed F-ratio Degrees of freedom for the test The “p-value” of the test Any post-hoc or planned comparison results “The mean score of Group A was 12, Group B was 25, and Group C was 27. A 1-way ANOVA was conducted and the results yielded a significant difference, F(2,25) = 5.67, p < Post hoc tests revealed that the differences between groups A and B and A and C were statistically reliable (respectively t(1) = 5.67, p < 0.05 & t(1) = 6.02, p <0.05). Groups B and C did not differ significantly from one another” 1 factor ANOVA

We covered much of this in our experimental design lecture
More than one factor Factors may be within or between Overall design may be entirely within, entirely between, or mixed Many F-ratios may be computed An F-ratio is computed to test the main effect of each factor An F-ratio is computed to test each of the potential interactions between the factors Factorial ANOVAs

Factorial ANOVAs Reporting your results The observed differences
Because there may be a lot of these, may present them in a table instead of directly in the text Kind of design e.g. “2 x 2 completely between factorial design” Computed F-ratios May see separate paragraphs for each factor, and for interactions Degrees of freedom for the test Each F-ratio will have its own set of df’s The “p-value” of the test May want to just say “all tests were tested with an alpha level of 0.05” Any post-hoc or planned comparison results Typically only the theoretically interesting comparisons are presented Factorial ANOVAs

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