Stats Flashcards
Standard deviation formula
sigma = sqrt[sum(x-xbar)^2/n-1]
Where n-1 = degrees of freedom
Standard deviation
Square root of the variance
T-tests
Compare 2 means to see if they are different from one another
Types of t-test
One-sample t-test
Two-sample t-test (unpaired, independent samples)
Paired-samples t-test (dependent samples, repeated measurements)
One-sample t-test
Tests the mean of a single group against a known mean
Two-sample t-test
Compares the means from 2 separate sets of independent and identically distributed samples from 2 populations
Paired-samples t-test
Compares means from the same group at different times
Used for comparing 2 measurements on the same item/person/thing
Welch’s t-test
An adaptation of Student’s t-test that is more reliable when the samples have unequal variances/unequal sample sizes
Still maintains the assumption of normality
Can be generalised to more than 2 samples which is more robust than ANOVA
Assumptions of a t-test
Data normally distributed/symmetrical
Data from multiple groups have the same variance
Data must be numeric and continuous
Data are independent (i.e. the data collection process is random)
How to calculate if data from multiple groups have the same variance
Take the lowest and highest standard deviation values - if the highest standard deviation is less than twice the lowest standard deviation, the variance can be said to be the same
Type I error
Concluding there is a difference when there isn’t
i.e. incorrectly rejecting the null hypothesis
Type II error
Concluding there isn’t a difference when there is
i.e. incorrectly accepting the null hypothesis
Family error rate
alpha
The probability that a test consisting of more than one comparison will incorrectly conclude that at least one of the observed differences is significantly different from the null hypothesis
Increases with each comparison
Non-parametric tests
Used when the data does not fit a normal distribution
Can be used for ranked data
Much wider applicability than parametric tests because they make fewer assumptions - this also means they are more robust - but they do have less power in cases where a parametric test would be more appropriate
BUT parametric tests can never be applied to data that is obviously non-parametric
Mann-Whitney U-test
= Wilcoxon rank-sum test
= non-parametric equivalent of the independent 2-samples t-test
Compares differences between 2 independent samples when the dependent variable is either ranked/ordinal, or is continuous but not normally distributed
Only valid for 2 groups
Can correct from multiple comparisons (family error rate)
Wilcoxon signed-rank test
= non-parametric equivalent of the paired-samples t-test
Used to compared 2 related/matched/dependent samples / repeated measurements on a single sample
Uses ordinal/ranked data
Spearman’s rank correlation coefficient
= non-parametric version of Pearson’s
= non-parametric test that measures the strength and direction of association between 2 ranked variables
i.e. measures the correlation
Data is ranked
One-way ANOVA
A technique for comparing the means of 3 or more groups of data
One-way ANOVA null hypothesis
Samples in all groups are drawn from populations with the same mean
F-statistic
Produced from one-way ANOVA
Calculated by dividing the variance of the means between the groups by the variance within the groups
If the group means are from populations with the same mean values, the variance between the group means should be lower than the variance of the samples, following the central limit theorem
When is F large?
When there is a larger difference between groups but little variance within groups
Larger F = probability that H0 is true decreases
Assumptions of ANOVA
Same as those for other parametric tests
Two-way ANOVA
A technique for comparing the means of 3 or more groups of data where 2 independent variables are considered
Can assess the effect of each independent variable on the dependent variable as well as if there is any interaction between the 2 independent variables
Repeated measures ANOVA
Equivalent of the one-way ANOVA but for related rather than independent groups
An extension of the paired-samples t-test
Detects ant overall difference between related means
Also called ‘within-subjects ANOVA’ / ‘ANOVA for correlated samples’
Kruskal-Wallis
= non-parametric equivalent of the one-way ANOVA
(also called ‘one-way ANOVA on ranks’ because it is a rank-based test)
Avoids assumption violations because non-parametric
Can compare 2 or more independent samples of equal or different sample sizes
Null hypothesis assumes the samples are form identical populations
Considered an extension of the Mann-Whitney U-test
Tukey’s test
Post-hoc test for ANOVA
i.e. if you have run an ANOVA and found significant results, you can run a Tukey’s test to find out which specific groups’ means are different
Protects from false positives
Same assumptions as those for parametric tests
An extension fo the t-test to multiple group comparisons
Dunnett’s correction
Post-hoc test for ANOVA
Similar to Tukey’s, except where Tukey’s compares every mean to every other mean, Dunnett’s compares every mean to a control test
Should only be used if there is a control group (if there isn’t, use Tukey’s)
Acts in a similar way to a t-test because it compares 2 groups
Bonferroni correction
Post-hoc correction
Limits the possibility of getting a statistically significant result when testing multiple hypotheses
Used when performing many dependent or independent statistical tests at the same time (when performing many simultaneous test, the probability of a significant result increases with each test run)
Sets the significance cut off at alpha/n, i.e. if you run 20 simultaneous tests at alpha=0.05, the correction would be 0.0025
Suffers from a loss of power - i.e. it sometimes over-corrects for type I errors, leading to type II tests
SNK method
= Newman-Keuls / Student-Newman-Keuls method
Post-hoc test
Again used to identify sample means that have been shown to be significantly different from one another following an ANOVA
Very similar to Tukey’s but uses different critical values
More like to commit type I errors than Tukey’s test (more powerful but less conservative than Tukey’s)
What does the ‘power’ of a statistical test refer to?
The probability that the test will reject a null hypothesis
