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Summary of Terms
Tuesday, 6 October, 2015

7:58 PM

Lecture 1

Datum: A single observation
Data: Recorded set of observations
Statistics: Summary descriptions of data

Model: Formal statements of regularities in data
- Also known as the rule used to account for the data
- The appropriateness of a model lies in its capacity to reproduce (or "account for") the
variability in the data and its ability to explain and predict
- Also referred to as a signal within the data
Lecture 2

Noise: Random fluctuations in measurements
- Considered to be the result of a multitude of small, but uncontrollable influences

Replicability: Concept of identical conditions yielding identical results
Strict replicability demands that precisely the same measurement be obtained on each repetition of
an observation under a given set of conditions
General replicability requires only that a repetitions of an investigation should lead to the same
general conclusion (follows the same model), but not necessarily identical numerical values (due to
noise in the data)
Lecture 3

Variables: Any attribute of objects, people or events that, within the context of a particular
investigation, can take on different values
Constants: Any attribute of objects, people or events that, within the context of a particular
investigation, has a fixed value

Quantitative variables: Variables whose values are in numbers
Categorical variables: Their value is the name (or symbol) that designates the category to which a
particular object, person or even belongs
...
Lesser variability, higher precision
Lecture 4

Experimental control: Corresponds to the common-sense idea that the conditions being compared
should only differ with respect to the factor under investigation
...
Ensures that any effect of inequality are non-systematic (purely by
chance)

Confounding variable: An uncontrolled factor that varies systematically with the predictor variable,
causing the results of an experiment to become ambiguous

Descriptive statistics: The use of graphs and summary statistics to capture the basic features of data
- Aids in communication, estimation and 'health check' of data
Outliers: Data points that are way out of line with the rest of the sample
Lecture 5

The correlation coefficient r: The index of the linear relationship between two variables
- Lies between +1
...
0 (perfect negative correlation)
○ 0
...
Is sensitive to outliers
Interquartile range: Difference between first and third quartile
Deviation scores: Difference between score and mean
...
Only negative scores are affected
(bias)
Sum of squares: Sum of squares of deviation scores
...
5x IQR below Q1 or above Q3
Linear Transformations:
Transforming Scores:

=

=

×

+

Lecture 8

Parameters of models: µ

Residuals: Added to account for variability in a model
...
Assumption that events are independent and mutually
exclusive
Binomial Distributions: Discrete probability distribution
-

( )=

(1 − )

Law of Large Numbers: As the number of observations increase, the observed provides an
increasingly accurate estimate of the probability to parameter values

Random Selection: A process which ensures that each element has an equal chance of being
selected
Random Sample: A sample selected from a population such that each possible sample of size n has
an equal chance of being selected

Central Limit Theorem: A variable that is the net result of a large number of independent outcomes
will have an approximately normal distribution
...
Probabilities of
particular values cannot be calculated; only probabilities of ranges of values can be calculated
...
g
...
Varies with degrees of freedom
...
This allows a
conclusion to be drawn even when the results are opposite to what was hypothesised
○ If one-tail is used, it must be explicitly written that it was used: t(20)=2
...
030 (1-tail)
- Reported as absolute values to 3dp
...
000" in MINITAB are to be written as ">
...
20

○ 0
...
80

(for independent groups design)
Small Effect

Medium Effect
Large Effect

(for match-pairs/related samples design)

Power
- Power, sample size and effect sizes are correlated
- Need two to estimate one
| =
...
50 ) =
○ (
- Tables available on MINITAB for
○ 1-sample z-test
○ 1-sample t-test
○ 2-sample t-test
○ Paired t-test
○ One-way ANOVA
Lecture 20

ANOVA (by R
...
Fisher)
- For designs with >2 conditions or >1 predictor variable
- Degrees of freedom:
○ Df Total = N-1
○ Df Model = k-1 (where k is the number of experimental conditions)
○ Df Error = Df total + Df model
- Model-fitting and sums of squares:
○ SSTotal: SS of residuals for Null Model
○ SSError or SSWithin: SS of residuals for Full Model
○ SSModel or SSBetween: Difference between SSs (represents variability between condition
means)
○ MS in ANOVA is the division of SS by df
- Test statistic: F (dfmodel, dferror)=
○ Is ratio of two variances (or mean squares)
○ If H0 is true, F=1
...
F<1
...
0 implies the presence of an
experimental effect
PSYC1104 Intro to Statistical Methods Page 7

experimental effect
- F-distribution is positively skewed, rejection regions varies with df
- Assumptions:
○ Normality
○ Homogeneity of Variance
- Relationship between F-statistic and t-statistic: F(1, v2) = t2(v2)

Lecture 21
The treatment effect (τ)
- τi is (µi - µ) is the difference between the condition mean and the overall mean
Lecture 22
Error rate per comparison (PC)
• Probability of making a Type 1 Error on any given comparison
• α=α', where α' is the error rate

Familywise error rate (FW)
• The probability that the conclusion drawn from the drawing of more than one comparison
(aka family of comparisons) will contain at least one Type 1 Error
• α=1-(1-α')c, where α' is the error rate and c is the number of comparisons

Controlling familywise error rate (Bonferroni Method)
• Using a more stringent α' for each comparison: divide overall significance level by number of
comparisons
α
○ α' =

○ Alternatively, multiply associated probabilities by number of comparisons and compare
with standard 5% significance level

Factorial ANOVA
• Extending ANOVA to designs with >2 predictor variables (aka factors)
• Eta-squared (η2): Used in place of R2, η2=

Main effects and interactions of factors
• Main effect: Difference between levels of a factor ignoring effects of other factors
• Interaction: Checking of factors are independent by looking at effect on one another
○ Additive effect: When factors are independent
○ Significant interaction: When factors influence each other
Lecture 23

Regression analysis (also: correlational analysis)
• Investigation of strength of relationships
• Construct an equation for predicting values of the response variable from values of predictor
variable

Fitting a regression line to data
• Regression line: | = ∗ + ∗
○
| is the mean of the population of Y values for a given value of X
○ a* is the intercept of the population regression line
○ b* is the slope of the population regression line (regression coefficient)
○ Best fitting line is one which minimises the sum of squares of the residuals, ie
...
Normal approximation to
binomial is used too
...
0 (absolute), the cell in question is a major contribution to the overall χ2 value

Lecture 27

Chi-squared test for a contingency table
1
...
Use the marginal proportions to calculate the expected proportions in each cell (under the null
hypothesis of independence)
3
...
Calculate the residuals by subtracting the expected frequencies from the observed frequencies
in each cell, (fo - fe) = e
5
...
In a two-way table, the degrees of freedom for
the chi-square value are obtained using the following formula: degrees of freedom = (R - 1) x
(C - 1) where R and C are the number of rows and columns
 
(1) =

a
...
e
...


(1,

 

e

= 50) = 12
...
001

Lecture 28

Measure of Association: Phi coefficient
•

=

 

• Size interpretation:
○
= 0
...
3 → Medium effect
○
= 0
...
Assign 1 and 0 to each level of the variables being tested
2
...
Compute Pearson's Product-Moment Correlation Coefficient (because phi is r but for binary
variables)

R x C Contingency Tables
• Null: the two bases of classification are independent
○ P(Help|0) = P(Help|1) = P(Help|2)
×

• Expected values:

• Chi-squared value: χ (2) = ∑  

(

)

= ∑  

• Tables can be collapsed if it makes theoretical sense

Measure for Association: Cramer's V
• Cramer's Phi (V):

=

 

(

, where k is the smaller of R and C

)

• Also the same thing as phi and r
• Size interpretation:

PSYC1104 Intro to Statistical Methods Page 10

• Size interpretation:

Smallest dimension of contingency table
2 (df=1)
3 (df=2)
4 (df=3)

Effect size
Small

Medium

Large


...
21


...
10

...
30

...
50

...
Original values are converted into ranks for both conditions
b
...
Smaller of the two test statistics is taken as the test statistic to report
○ Stat --> Non-parametric --> Mann-Whitney

Wilcoxon Signed-rank Test
• Hypotheses:
○ H0: P(The rank is positive) = 1/2
○ HA: P(The rank is positive) ≠ 1/2 (two-sided)
OR HA: P(The rank is positive) > 1/2
OR
○ H0: θD = 0
○ HA: θD ≠ 0
Computation:
a
...
If the difference is zero, the pair of scores is dropped from the analysis
i
...
The differences are ranked, ignoring the direction of difference
i
...
Sum the ranks with positive signs (T+) and negative signs (T-) separately
○ Compute the differences, then run: Stat --> Non-parametric --> 1-Sample Wilcoxon
Large Sample Approximation to the Non-Parametric Sampling Distribution
• As sample size increases, W and T tends to normality
PSYC1104 Intro to Statistical Methods Page 11

• As sample size increases, W and T tends to normality
• A z-score can be computed and used as the test statistic
○ Wilcoxon W Statistic (When or
>10):
( +
+ 1)
=
2
(

 

=

−

=

+
12

+ 1)

○ Wilcoxon T Statistic (When N>15):
( + 1)
=
4
=

=

 

−

(N + 1)(2 + 1)
24

PSYC1104 Intro to Statistical Methods Page 12



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