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Statistics revision
Measurements
Measurement = the assignment of numbers to objects, events or latent constructs
The main approach of psychometric measurements involves applying interviews, questionnaires and
tests (= instruments)
...
Error is the main thing that impacts on repeatability and reproducibility
...

Variables and types of data
Observed variable = directly observable and can be measured easily
...
Instead, it is inferred from
variables (items) than can be observed
...
This
comes under continuous scale
Qualitative = Categorical variables measure a quality or characteristics on each individual or object
Two measurement scales for categorical variables:
-

A nominal scale gives unordered values/categories to the variables,
An ordinal scale assigns ordered categories
Quantitative = measure a numerical quantity or amount on each individual or object
...
5, 6, 6
...

It aims to give a typical sense of the characteristics under investigation
...
The data itself is considered a one-way table
...

Relative frequencies (proportions) in the target population are referred to as probabilities
...
Mean (highly influenced by extreme points)
2
...
Mode (appropriate for nominal or ordinal scales)
4
...
It allows us to look at a
“typical” or “average” case
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Aims to give a typical sense of the characteristics under investigation
Measures of centre are descriptive statistics that give an idea about the location of a set of
measurements

-

-

Two most commonly used measures of centre are:
o The mean or average – is defined as the sum of a set of measurements divided by
the number of measurements
o The median – is a value that falls in the middle position when the measurements are
ordered from smallest to largest (for even number of measurements, there will be
two middle values, and the median is estimated as the average of the two values)
...
If they are distinctly different you will need to
look at transforming the data

Measures of spread/variability:
-

-

-

-

After you look at what is typical of the data, you need to look at if this is representative of
the true population
1
...
Standard deviation (interval scale, cannot be used in skewed distributions)
3
...
Variance (used for interval scale)
Mean = standard deviation
Median = range/IQR
The range is associated with the median; it is the difference between the lowest value and
the highest value you have in your data
...
This measure although simple to calculate is limited by the influence from
extreme values it uses in its calculation
The mean deviation is a means of looking at how different our individual values are from the
mean but is not commonly used in statistical analysis
...

The standard deviation is similar to the mean deviation but instead of measuring the change
of values from the mean it tries to capture the average distance of each value from the
average by using the squares of these changes from the average
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Another measure which is related to the SD is the standard error
...
It is only valid to use when your
sample of data has been randomly generated
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64

Graphical representations of data
Rules for graphs:
-

Give the graph an appropriate title;
Clearly identify the categories or values of the variable;
Indicate the units of measurement;
Indicate the total number of cases;
Indicate the source of the data (indicated in the figure legend)
...

They can be used to describe the summary data (descriptive stats) or to display the results of a
statistical analysis (e
...
confidence intervals)
Histograms
-

Provide a visual impression about the centre, spread and shape (symmetric or skewed) of a
distribution
They can be used to compare distributions

Box and whisker plots
-

-

-

Useful for comparing groups or variables
The box of a box plot contains 50% of the
data, from lower quartile to the upper
quartile
Multiple Box-whiskers are very useful in
visually comparing groups of different
variables
...


Note where the mean sits! Because the mean is calculated from every value in the
distribution it is influenced by extreme scores or outliers, so when the distribution is skewed
the mean may give either a lower or higher value than is the true average
...


The population and sample
Population and Parameter (population mean)  Sample and Statistic (Sample mean)
In most statistical problems, a specified number of measurements or data, i
...
, sample is taken from
a much larger body of measurements – called the population
A population is any entire collection of people, animals, plants or things from which we may collect
data
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This is often best achieved by random sampling
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Usually we don’t know this
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Usually denoted by Roman letters: 𝑥̅  sample mean p  sample proportion
s  sample standard deviation

Sampling distribution
A sampling distribution is the shape of the data
...

An important concept in statistics is the sampling distribution
Sampling distribution = the (probability) distribution of a statistic over repeated sampling
A statistic may refer to anything calculated from a sample
Although most statistical analyses are based on a single sample, statistical theory can inform us
about the behaviour (distribution, variability etc
...
g
...
Standard
error tells us how variable this estimate is – it is the standard deviation of the sampling distribution
...
Take all the means and work
out their standard deviation and just call it the standard error
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Estimation
Estimation = the process by which sample data are used to estimate the unknown value of a
population parameter
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The motivation of an interval estimate is to give a plausible range to the population parameter,
rather than estimating it by a single value
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Bias and standard error of estimators
The sampling distribution can inform us about the average estimate values and their variability
...
5, equal to the true parameter (=0
...


The standard deviation of the sample statistic over repeated sampling is called its “Standard Error
(SE)” and tells us something about variability
...
Precision increases with the sample size n
...
(Which is unknown!)
A sample statistic that has this property is called a consistent estimator
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e
...

Confidence intervals
A confidence interval is to give a range of values in which the true value is believed to lie
Confidence intervals are required in any publications
...
96×SE
Upper limit = estimate + 1
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96 comes from the z normal tables
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If we define confidence intervals in this way for repeated samples, then 95% of them will contain the
true value of the population parameter (, or )
...

The more confidence is required, i
...
the larger the confidence level, the wider the confidence
interval (CI)
...
e
...
95) using the mean +/- 1
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For a 90% CI, we would use +/- 1
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These values of 1
...
65 come from the standard normal distribution (sometimes called the zdistribution) and apply when the sample is large
...
g
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23 standard errors for a 95% CI
...

Confidence intervals for probabilities can be constructed through two methods:

1
...

2
...

The CI properties hold only for large sample sizes
...

Where available exact Cis are preferable to asymptotic Cis (although this doesn’t matter for larger
samples)
An approximate 95% CI for the probability  is given by: [𝑝 − 1
...
96√

𝑝(1−𝑝)
𝑛

]

where p is the observed relative frequency of the category in a sample of size n
The approximation is valid if both np > 5 and n(1-p) > 5
For rare events this may require large sample sizes
...
e
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It is the probability of getting the observed data under the assumption that the null hypothesis is
true
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How exactly depends on the type of the alternative hypothesis:
-

One-sided alternative hypothesis → respective tail (right or left) of the sampling distribution
is used
...


We choose a right tailed test (right rejection region) when we are confident that negative difference
can happen only by chance
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When in doubt about the direction of the difference, choose a two-tailed test
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This critical value defines the rejection region
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So, either 5% split either side or 5% on
one side
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It is best to avoid saying that we “accept the null hypothesis” because a test only controls the
probability of the error that we might have made when taking the decision to reject (type-I error)
...

Sampling distributions (repeated sampling and calculating a mean and SE as a population estimate)
of statistics are used to perform tests concerning the values of population parameters
...

2
...

4
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The null hypothesis, denoted by H0
The alternative hypothesis, denoted by H1
The test statistic and its p-value (which test and what significance level)
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The conclusion/decision
...
It reflects the
investigator’s belief about the unknown parameters
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Finding an effect can be overturned through collection of further data
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H1:  > 0 (mean is greater than 0, one-sided alternative hypothesis; the investigator knows
that some values are not possible)
2
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H1:   0 (two-sided alternative hypothesis, i
...
 > 0 or  < 0, usually a more defensible
alternative)
Errors in hypothesis testing:
-

-

-

Type-I error:
o A Type-I error is the error of rejecting the null hypothesis when it is true
...

o The probability of making a type-I error is denoted by  (also called the significance
level)
...

o This is a false negative result
...

The power of a statistical test is given by (1-), which is the probability of making a correct
decision by rejecting a false hypothesis
...


-

BETA should be lower than ALPHA if the cost of the trial is higher than its benefit to
knowledge (i
...
looking at a drug and its side effects)
...
e
...

This particular one is called the one-sample t-test statistic
...
f
...
)
Reporting a one sample t test: t (490) = -7
...
001
Testing a one sample proportion
When you look at the probability of a category against a pre-specified value
Or it might be when you want to see if a group is under-representative
...
This is an alternative test allowing tests of
proportions rather than means
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Assumption is that each category will have more than a value of 5 (i
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3 males and 200 females)
If the sample is large enough you can use a binomial test which is an exact test
Asymptotic or a Monte-Carlo test when the sample is small
...


Independent samples t-test
Inferences assume both groups are samples from normal distributions
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95% Confidence interval for a difference between means: [d – 2SE(d), d + 2SE(d)]
Test the null hypothesis to investigate whether the means are the same: 1= 2
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Levenes test
Levenes test is an inferential statistic used to assess the assumption of equality of variances
(homogeneity of variance or homoscedasticity) for a variable calculated for two or more groups
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Value is greater than
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A value greater than
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That the scores in one condition do not vary too much more
than the scores in your second condition
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Two variances are not statistically different so the variances are equal
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Value is less than or equal to
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A value less than
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That the scores in one condition vary much more than the scores
in your second condition
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This is a bad thing, but SPSS takes this into account by giving you
slightly different results in the second row
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Mann-Whitney U-test
Used as the non-parametric alternative for two independent samples or generally for ordinal data
...
Ranks subjects within the combined sample
2
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Constructs a test statistic that is a function of the sum of the ranks within the groups (often
denoted “u” or “w”)
4
...

Analyse  non-parametric  legacy dialogues  2 independent samples
When there is ordinal data you use monte-Carlo with the Mann-Whitney
There is also no mean with ordinal data
Comparing binary outcomes
For binary outcomes, the distribution in each sample is characterised simply by the proportion of “1”
s in each sample
Dummy variable when if you weren’t a “1” you automatically become a “0”
The sample proportions provide unbiased estimators of the population proportions
...
Confidence intervals can also be derived (no details here)
...
This amounts to testing the null hypotheses (all equivalent)
Equal proportions of “1s” in the two populations (RR=1)
In the remainder, we introduce two tests for such hypotheses
-

The chi-squared test
Fisher’s exact test

Relative risk and odds ratio
A contingency table summarizes the frequency distribution of each of two categorical variables as
well as the association between two categorical variables
Each cell of a two-way table contains the frequency at which combination of its row and column
categories occurred, the row and column totals represent the distributions of the variables singly

Two way tables allow the calculation of
risk and odds
...

You can display them as a percentage of
the total population
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374
So, there is a 37% less risk of males suffering from Malaise
Odds ratio: OR =

𝑇ℎ𝑒 𝑐ℎ𝑎𝑛𝑐𝑒 𝑜𝑓 𝑀𝑎𝑙𝑒𝑠 𝑠𝑢𝑓𝑓𝑒𝑟𝑖𝑛𝑔 𝑀𝑎𝑙𝑎𝑖𝑠𝑒 𝑐𝑜𝑚𝑝𝑎𝑟𝑒𝑑 𝑡𝑜 𝑛𝑜𝑡 𝑠𝑢𝑓𝑓𝑒𝑟𝑖𝑛𝑔 𝑀𝑎𝑙𝑎𝑖𝑠𝑒
𝑇ℎ𝑒 𝑐ℎ𝑎𝑛𝑐𝑒 𝑜𝑓 𝐹𝑒𝑚𝑎𝑙𝑒𝑠 𝑠𝑢𝑓𝑓𝑒𝑟𝑖𝑛𝑔 𝑓𝑟𝑜𝑚 𝑀𝑎𝑙𝑎𝑖𝑠𝑒 𝑐𝑜𝑚𝑝𝑎𝑟𝑒𝑑 𝑡𝑜 𝑛𝑜𝑡 𝑠𝑢𝑓𝑓𝑒𝑟𝑖𝑛𝑔 𝑀𝑎𝑙𝑎𝑖𝑠𝑒
26

72

26 ×437

= 465 ÷ 437 = 465× 72 = 0
...
g
...
5 then the odds of males suffering malaise is 1
...

An odds ratio is a special type of ratio, one in which the numerator and denominator sum to one
...

We may wish to check whether there is any association between the row and column variable
...

Statistical independence means that knowing the value of one variable does not provide any
information regarding the distribution of the other, or formally for any two categories A and B:

Prob  A and B  Pr ob  A   Pr ob B 

For statistical test of independence, a Chi Squared test is used
...

The idea behind this test is to compare the observed frequencies with the frequencies that would be
expected if the null hypothesis of no association / statistical independence were true
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(This is what is usually meant by “the” chi-squared test
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Sum of: (x (value) – e (expected value))2/e
The greater the deviations the greater the chi value

Analyse  descriptives  crosstabs  statistics  tick Chi-Squared
Cell tab – can include percentages and counts
Exact tab – depending on sample, Monte Carlo if not enough sample or Asymptotic
SPSS automatically calculates:
-

The OR of the event in the first row comparing the first column with the second = the OR of
the first column comparing the first row with the second
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The RR of the event in the second column comparing the first row with the second
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Number is
greater than 1 so males are less likely to have malaise
...


Fisher’s exact test
While well-known, the chi-squared test of independence can only be applied when the expected cell
counts are large enough
...

-

Results can take a while for large tables due to Monte-Carlo simulation
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Parametric vs non-parametric
-

-

Advantage of nonparametric tests:
o do not require any distributional assumptions
o can be used for ordinal data
Advantage of parametric tests:
o when possible more powerful than nonparametric equivalents when distributional
assumptions are fulfilled (sometimes this can be achieved by data transformation)
o concepts can be expanded to address more complex research questions:

Fixing group sizes by design
A survey does not determine the sizes of the groups of exposed or non-exposed subjects (E-groups),
nor the sizes of the groups of cases or non-cases (O-groups)
E
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in the alcohol misuse and gender data set the gender group sizes and the offence type group
sizes were random
A prospective study fixes the sizes of the E-groups
E
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children of 1000 women with and 1000 without birth complications are followed up over time to
determine whether they develop Antisocial Personality Disorder (ASPD)
A retrospective study fixes the sizes of the O-groups
E
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Exposure to childhood trauma is determined for 100 children with ASPD and 100 children
without ASPD
Surveys and prospective studies do not predetermine the distribution of the outcome
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g
...

The statistical methods introduced last week can be used to evaluate any effect size that may be of
interest
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g
...
g
...


Consider the three most common research designs:
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Survey (cross-sectional study)
Prospective study (cohort study)
Retrospective study (case control study)

In a prospective study, non-exposed individuals may be matched to exposed individuals using
matching characteristics
...

In a survey, could select subset of non-exposed that match exposed subjects (you could also match
non-cases to cases but not recommended)
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A retrospective study can estimate a particular type of odds ratio: The OR comparing the odds of
being exposed between cases and controls
...

OR= #discordant pairs with only case exposed / #discordant pairs with only control exposed
An RR cannot be directly estimated from a case control study because such mathematical
equivalence does not hold for the RR
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However, an RR can be approximated by the OR of the outcome is rare (say less than 5%)
Matching
Matching is part of the (statistical) design of a study, it needs to be acknowledged in the statistical
analysis
...

o Inferences based on paired data can be more efficient than inferences based on
independent samples if the pair members are very similar w
...
t
...

To eliminate or minimize confounding
o We want to ensure that the E-O association that we observed in the sample cannot
be accounted for by the influence of confounding variables
...

Analyse  compare means  paired samples t-test
Wilcoxen signed ranks test
This calculates the differences within the pairs, ranks the absolute differences (irrespective of + or -),
reassigns + and – to the rank based on what the sign was initially and compares the ranks of pairs
with positive differences with those with negative differences
...

Analyse  non-parametric  legacy dialogues  Two-related samples tests  tick Wilcoxen
McNemar test
This is used when there is a binary outcome (for example if they are older than 40 or not and seeing
if wives and husbands differ)
...
g
...
e
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When there is a significant E-O association the off-diagonal cells in the 2x2 table can be looked at to
understand its direction
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Y is dependent on another variable
...
This is the value that y takes when x is zero
...
This is a constant value, when this is 0 what is the residual impact on y
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This determines the change in y when x changes by 1 unit
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Every regression line has an element of error
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The relationship between variables is a linear function:

E is the error
(discrepancy between
the expected and
observed)

Using the term ‘population’ as we are estimating what it would be in the population from the
sample
...

A scatterplot is the first point of call before conducting a regression
...

Graphical display of a bivariate relationship
Distributions for two continuous univariate variables can be compared using univariate histograms
and boxplots
...
This is achieved by a
scatter plot of individual data points
...
You can also include a confidence interval by double clicking on the line
...

Maximum Likelihood (ML) – chooses the line under which the observed data was most likely
to have occurred
...

Statistical software provides estimates of the intercept and slope, together known as regression
coefficients
...
626kg difference in weight
...

But what about the intercept (  0 ) what does -46
...

Regression is only valid for the scope of data that you have collected
In addition to the “point” estimates of the regression coefficients we also get estimates of how
precisely we have estimated them – the standard error (Std
...
96SE(B), B + 1
...
This is also given in the output
...


The test statistic is t=B/SE(B)
The value of the test statistic can be compared with a t-distribution with n-2 degrees of freedom to
obtain a p-value
...

The slope coefficient simply measures the group difference in means (remember: slope
measures predicted change in y when x changes by one unit=switches groups)
This is the same as an independent samples t-test (and a t-test can be thought of a statistical
model)
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Changing the switching the codes between male and female would give a positive slope
(mean difference)
...

This is when there is more than 1 variable
1 = present and 0 = absent
Categorical variables need to be recoded into dummy variables before including them in a
regression model
...

The non-coded category is treated as a reference category
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Assumptions required for inference:
-

The observations on y are independent
...
This is the equivalent of assuming that the
within group variances are equal when doing a two-samples t-test
...
Here 27% variability in
weight is explained by height
...

-

The model summary reports R2 and an adjusted version of R-squared
...


Pearson correlation r
Usually this is conducted at the start, it tells you potentially how strong the relationship could be
...

The Pearson product moment correlation r between two continuous variables measures the
strength of the linear relationship
...
dev
...

This concept of correlation (standardised directional association) is less sensitive to extreme
influential points
...

The violent crime versus poverty data has one clearly highly influential point that increases the
strength of (directional) association
-

Pearson correlation 0
...
39

Confounders and adjusting for them
Statistical bias is the difference between the expected value and the true value
...

Bias = E(B) – B (error of Beta – Beta)
One source of bias is confounding
...
Confounds have a relationship with both E and O
...

Confounders can be dealt with at the design stage or at the analysis stage depending on when
awareness of a confound becomes apparent
...

In the analysis stage this is through statistical modelling (multiple regression analysis)
...


Outcome (y) must be continuous, predictor variables are continuous or categorical
...

Multiple linear regression model: y   0  1 x1     k xk  
Instead of one predictor you have multiple in the regression
...
Because of this we can
say that it is a confounder
...

Using multiple regression:
-

Multiple regression framework is a natural and practical way of adjusting for confounders

-

To adjust the E-O association for C, all we need to do is include C in the regression model of
O on E as an additional predictor, which will automatically adjust the E-O association for C
...
e
...


After adjusting for education poverty is now not significant, indicating that education was
causing the previous simple linear regression significance
...

A rule of thumb for multiple regression is to ensure that there are more than 10 observations (data
points) per predictor variable
...


Both education and being a single parent are potential
confounders to the poverty-crime association
...


Poverty-crime association does not exist (β = -2
...
844) in the adjusted model, but it
appears, the percentage of single parents is the most important determinant of the crime rate (β =
124
...
001)
...

When you have, multiple predictors use adjusted R2 as it tells us whether adding factors improves
the model
...

Types of multiple regression – stepwise (putting variables in and then taking them out depending on
what is best), forward, or backward
...
6%
R-squared and explained variance:
-

R-squared indicated that x explains Z amount of the variance in y
R2 increases for each additional predictor variable included in the model
If single and poverty were independent, we would expect them to explain (42+14) = 56%
when considered jointly
In practice, they explain only 43% - just 1% more than that explained by single alone
This implies, given the effect of single has been taken into account, poverty explains only 1%
extra

Adjusted r-squared:
-

-

Adjusted R2 is a modified version of R2 that adjusts for the number of predictor variables in
the model
R2 always increases for each additional predictor, but adjusted R2 can either increase or
decrease
Adjusted R2 increases only if the contribution of an additional predictor is sufficient to
compensate for the loss of degrees of freedom due to inclusion of an extra predictor
variable in the model
Adjusted R2 is considered to be a better indicator for model selection

Assumptions of regression (linear and multiple) – EXAM IMPORTANT
Normality: the error terms (residuals) are normally distributed (save the residuals and produce a
histogram, normal probability plot of residuals, sometimes called q-q plot – SPSS can do this for you
if you click the button)
Variance Homogeneity: The error terms have the same variance irrespective of the values of X (i
...
,
variance does not depend on X) (plot of standardised residuals vs
...
Click on the plot tab, tick histogram and normal
probability plot
...
Z is residual, SD is standardised
Confounding vs
...

They are both modelled in the same way in a regression equation
...

In SPSS, they are modelled in the exact same way
...

A mediator of the causal effect of exposure on outcome is a variable on the causal pathway from E
to O
...

In non-mediated models the total effect of exposure on outcome is denoted by path c’
...

For a linear regression with a continuous mediator, indirect effect can be obtained by calculating the
product of path a and path b (multiplying the pathways)
In regression type models, total effect is the sum of the direct and indirect effects: C = c’ + a*b

M
b=1
...
83

c = 6
...
00
E

O

Baron and Kenny Steps
Baron and Kenny (1986) discussed four steps to establish mediation:
1
...
Test path a (E  M)
Show that the causal variable is associated with the mediator
3
...
Test path c’ (E, controlling for M  O)
For complete mediation this should be not significant
For partial mediation c’ is smaller than c in absolute value
If indirect effect is the same as the total effect, we have complete mediation
...
e
...


Complete and partial mediation
Complete mediation would occur if the mediator accounts for all of the total effect (c’ = total effect
c)
Partial mediation would occur if the mediating variable accounts for some but not all the total effect
(and E is still significant in the model)
...

Methods for testing the indirect effect formally
The next step in assessing the indirect effect is by seeing whether a*b is significant
...
Sobel who originally proposed this test
Sobel test works well only in large samples
...

Sobel:
-

Sobel test is based on an approximate z-statistic, given by

𝑧=
-

-

𝑎𝑏
𝑆𝐸(𝑎𝑏)

SE(ab) denotes the standard error of the estimated direct effect, given by

𝑆𝐸(𝑎𝑏) = √𝑎2 𝑆 2 + 𝑏 2 𝑆 2
𝑎
𝑏
Where, 𝑆 𝑎 and 𝑆 𝑏 are the SE of a and b respectively
Limitations:
o Sobel test is based on normal approximation (z-test)
o Sampling distribution of ab is highly skewed
o Large values of ab are more variable than the smaller values
o This may lower the statistical power of the Sobel test

Bootstrapping:
-

Bootstrapping requires taking a large number of samples (with replacement) from the
original dataset
Indirect effect (ab) is estimated for each of the bootstrap samples
These bootstrap estimates are used to form a non-parametric distribution of the indirect
effect

Assumptions of mediation
Mediator (M) is a causal effect of the exposure (E)
The exposure(E) and mediator (M) do not interact to cause the outcome (O)
No omitted variables
All common causes of M and O, E and M, and E and O measured and controlled



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