What is the purpose of performing a linear regression analysis?
To identify potential outliers in the data
To fit the data to a model that defines y as a function or 2 or more variables
To determine the dependence of a dependent variable on a predictor/independent variable
To perform multiple comparisons whilst controlling overall type 1 error rate
To derive robust confidence intervals
Which axis does the dependent variable go on?
y
x
What does the mean of the x and y values give you in a linear regression analysis?
The size of the force which the points exert on the line of best fit
The leverage of those data points
The fit and slope of the model
The centre of gravity and pivot point of the data
What does the R-squared value represent?
How well the model fits the data (0 - 1)
The slope coefficient
The distribution of the residuals
The level of multicolinearity in the model
What does an R-squared value of 0.068 and a slope coefficient (b1) value of 0.12 mean?
The model can explain 68% of the data and for every unit of independent variable, the dependent variable goes up 12 units
The fit of the model to the data is 0.12% and the influence that the data points have on the model is 0.68%
The data points have an influence of 68% on the model and 12% on the outcome
The model can explain 6.8% of the data and for every unit of independent variable, the dependent variable goes up 0.12 units
In order to identify potential outliers:
Standardised residual >2 is worth checking, if more than 5% of the residuals >2 may indicate model is a poor fit
Standardised residual >3 is worth checking, if more than 5% of the residuals >2 may indicate that the model is a poor fit
Standardised residual >2.5 is worth checking, if more than 5% of the residuals >2 may indicate that the model is a poor fit
Standardised residual >3 is worth checking, if more than 10% of the residuals >2 may indicate that the model is a poor fit
What does Cook's distance tell us when performing model diagnostics to see if the regression model is stable or biased by a few cases?
influence of data point on predicted values (0 = no influence, 1 = complete influence)
standardised measures of how much each element of the model would change if data point was removed (values >1 = substantial influence)
how susceptible the mean is to being biased by the outliers present in the data
measure of overall influence of each individual data point on the overall model (>1 = concern)
What does the Leverage value tell us when performing model diagnostics to see if the regression model is stable or biased by a few cases?
standardised measures of how much each element of the model would change if data point was removed (values > 1 = substantial influence)
measure of overall influence of each individual data point on the overall model (> 1 = concern)
precisely how large the standardised residuals are
With regard to model diagnostics, what do the DFFit and DFBeta values tell us about the data model?
they are standardised measures of how much each element of the model would change if that data point was removed (values > 1 = substantial influence)
they indicate the influence of that data point on predicted values (0 = no influence, 1 = complete influence)
whether or not the standardised residuals are worth checking and if they indicate that the model is a poor fit
they summarise the equation: 2(k+1)/n where k = number of predictors and n = number of data points
With regard to the model diagnostic called the Leverage value, what defines whether or not the data point is worth investigating?
if >2(k+1)/n where k = number of predictors (2 for simple linear regression) and n = number of data points
if >2(k+1)/n where k = number of predictors (1 for simple linear regression) and n = number of data points
if >2(K+1)/n where k = number of data points and n = number of predictors (1 for simple linear regression)
if >n(k+1)/2 where k = number of predictors (1 for simple linear regression) and n = number of data points
if >2(n+1)/k where k = number of predictors (1 for simple linear regression) and n = number of data points
Multiple linear regression does what?
fits the data to a model that defines y as a function of 2 or more variables - determines the effect of an independent variable on the dependent variable taking account of other variables
provides an analysis of variance and determines if an interaction is present in the data
determines the dependence of a dependent variable on a predictor/independent variable and allows outliers to be identified from x, y plot or from standardised residual plot
With regard to multiple linear regression, what is the correct form of the equation for the model which is fitted? (all of the numbers are technically subscript)
y = b0 + b1x2 + b2x1
y = b0 + b1x1 + b2x2 +....
y = b0 + b1 + b2x
y = b0 + b1x1 + b2x2
What does the F-ratio represent?
the average variability due to the model divided by the average variability due to the residuals
the unexplained variability divided by the variability due to the model
the signal to noise ratio multiplied by the number of data points
the variance in the model divided by the R-squared value
With regard to multiple linear regression, whenever you fit a predictor variable, that takes up...
one slope parameter
two degrees of freedom
one degree of freedom
one R-squared value
As colinearity increases what effect does this have?
standard errors of b coefficients decrease therefore confidence increases
limits F-ratio value and variance inflation factor
coefficients become stable
standard errors of b coefficients increase and therefore confidence decreases
How do you interpret the variance inflation factor (VIF) when assessing multicolinearity?
A VIF > 5 or an avereage VIF > 2 is problematic
A VIF > 10 or an average VIF > 1 is problematic
A VIF > 2 or an average VIF > 1 is problematic
A VIF > 10 or an average VIF > 2 is problematic
How do you interpret the tolerance factor when assessing multicolinearity?
< 5 is problematic
< 10 is problematic
< 2 is problematic
< 0.1 is problematic
< 1 is problematic
When does multicolinearity truly pose a problem?
when predicting y using the multiple regression equation
when you want to look inside the model at the effect of individual predictors
when you want to perform separate correlations for each x variable
when you want to quantify the relationship between an independent and dependent variable
How do you help solve the problem of multicolinearity?
always take a colinear variable out
combine predictors into a single predictor (as long as it makes biological sense)
rely on automatic variable selection
remove all outliers
With regard to hierarchical multiple regression, what value do you use when comparing new model to previous model?
F-ratio
Cook's distance
R-squared
variance inflation factor
tolerance factor
For multiple linear regression assumptions, what must the variables be?
dependent variables = quantitative or categorical predictor variable = qualitative and continuous
dependent variables = qualitative predictor variable = continuous
dependent variables = qualitative and continuous predictor variable = quantitative or categorical
dependent variables = continuous or categorical predictor variable = quantitative or categorical
When considering multiple linear regression assumptions, how do you assess the independence of the residuals?
assess the DFFit and DFBeta values
use the Welch's test
use Gabriel's test
use the Durbin-Watson test
For multiple linear regression, how large should the sample size be?
10 times the number of predictors tested
5 times the number of predictors tested
at least 30
at least 40
What would an interaction among predictors look like in the form of an equation?
effect of height + effect of weight = overall effect on SBP
effect of height + overall effect on SBP = effect of weight
effect of height x effect of weight = overall effect on SBP
What is simple linear regression equal to?
paired t-test
unpaired t-test
unpaired, two-tailed t-test
paired, one-tailed t-test
A one-way anova is the same as what?
simple linear regression
multiple regression
What does a one-way ANOVA do?
analyses how much of the overall variance can be explained by variation between group means compared to the unexplained variation within a group
fits data to a model that defines y as a function of 2 or more variables
performs separate correlations for each x variable
determines the dependence of a dependent variable on a predictor/independent variable
What does the total variability equal?
total squares divided by the degrees of freedom
the F-ratio
the difference between each individual data point and the overall mean
error mean squares divided by degrees of freedom
The F-ratio is:
higher the larger the difference of the group means from the overall mean and smaller the amount of random variability
lower the larger the difference of the group means from the overall mean and smaller the amount of random variability
higher the larger the difference of the group means from the overall mean and larger the amount of random variability
higher the smaller the difference of the group means from the overall mean and smaller the amount of random variability
When is the ANOVA most robust to deviations from normality and equality of variance?
when effect size is large
when the F-ratio is high
when the degrees of freedom are greater than 10
when the group sizes are equal
If group sizes are unequal and equality of variance is not met then which correction do you use?
Games-Howell's
Durbin-Watson's
Gabriel's
Tukey's
Welch's
What are post-hoc tests used for?
performing multiple comparisons whilst controlling overall type 2 error rate
performing multiple comparisons whilst controlling overall type 1 error rate
when there is a specific hypothesis to be tested
when group size is not equal
You use Tukey's test when which of the following is true? (multiple correct answers)
sample sizes are unequal
sample sizes are equal
you require good trade-off between type 1 and type 2 errors
you are interested in comparing all groups vs a single control group
you wish to cut down on the number of comparisons that you make
When would you use Bonferroni as a post-hoc test? (multiple correct answers)
when you don't need a high level of confidence
when you aren't performing multiple comparisons
when you require a conservative test
when you need a high level of confidence
when sample sizes are equal
When would you use Dunnet's as a post-hoc test? (multiple correct answers)
when interested in comparing all groups versus a single control group
when you want to cut down comparisons
when you want a good trade-off between type 1 and type 2 errors
Separate, unpaired t-tests to do comparisons will increase your chance of getting what?
a false -ve
a type 2 error
a false +ve
biased data
If you have a sample which has an n number of 10 and a sample with an n number of 12, which post hoc test should you use?
Hochberg's GT2
Games-Howell
Tukey
If there is any doubt about equality of variance then which post-hoc test should you use?
Sidak
Complete this statement relating to planned contrasts: Always _________________________ than number of groups
ten times the number of contrasts
more contrasts
one more contrast
two times the number of contrasts
one fewer contrast
When doing orthogonal contrasts, the contrasts are independent so you can... (multiple correct answers)
enter weights for most of the variables
trust p-values as you aren't inflating the type 1 error rate
ignore the F-ratio value and R-squared value
not worry about doing any corrections for multiple comparisons
"tests for trends in the data, which cannot be obtained directly using post-hoc tests, when there is a logical order to the groups entered" To what is this statement referring to?
Planned contrasts
Orthogonal contrasts
Polynomial contrasts
An independent factorial ANOVA does what?
each level of one factor is tested against at least one level of the other
performs multiple comparisons whilst controlling overall type 1 error rate
divides total variability in the data set into different sources
each level of one factor is tested at each level of the other
Sidak is the best correction for what?
independent ANOVA
one-way ANOVA
repeated measures ANOVA
multiple linear regression
Standard contrasts and post hoc tests are only available to examine main effects and are therefore most useful when:
there is no unnecessary variability in the data
there is no interaction
the sphericity assumption is met
group sizes are equal
A p value of less than 0.5 means that....
there is a less than 0.5% chance of committing a type 1error
there is a less than 5% chance of committing a type 2 error
there is a less than 5% chance of committing a type 1 error
there is a less than 0.5% chance of committing a type 2 error
Standard error the proportion equals....
Power =
1 - type 1 error rate
1 - type 2 error rate
1 - (type 1 + type 2 error rate)
type 2 error rate - type 1 error rate
Power can be increased by....
increasing effect size. decreasing random variation. decreasing sample size.
increasing effect size. increasing random variation. increasing sample size
decreasing effect size. increasing random variation. increasing sample size
increasing effect size. decreasing random variation. increasing sample size
This gives you a standardised effect size for a difference between means, what is it called?
Welch's correction
Cohen's d
Games-Howell test
Sidak correction
standard error of the proportion
how do you calculate expected frequency?
(row total + column total)/overall total
(row total - column total)/overall total
(row total x column total)/overall total
How do you calculate degrees of freedom from a contingency table?
df = (rows - 1) x (columns -1)
df = (rows + 1) x (columns +1)
df = (rows - 1) / (columns -1)
df = (rows x 2) + (columns x 2)
With regard to categorical data - what must be satisfied in order for the analysis to be reliable?
The assumption that at least 50% of expected frequency must be more than or equal to 5
Dunnet's test
The assumption that at least 80% of expected frequency must be more than or equal to 5
The same assumptions as multiple linear regression
Which graph indicates an interaction?
What does simple effects analysis do? (multiple correct answers)
probes where a certain effect is happening
performs an ANOVA to allow you to reject/accept a null hypothesis
analyses the differences between levels of one variable
performs multiple comparisons whilst controlling overall type 2 error rate
Which multiple comparison correction should you choose after simple effects analysis in order to control the overall type 1 error rate?
Bonferroni
LSD
Repeated measures ANOVA requires the data to have/be:
independent
not independent
naturally paired
sorted into even group sizes
What is the definition of sphericity?
“noise” in the relationship between the independent variables and the dependent variable is the same across all values of the independent variables
equality of differences between linked values in each group
well-modeled by a normal distribution and likely for a random variable underlying the data set to be normally distributed
residuals are (roughly) normal and (approximately) independently distributed with a mean of 0 and some constant variance
Which test provides a fix for sphericity?
Gabriel's test
Mauchy's test
How can we adjust the degrees of freedom and change the significance level associated with the F-statistic?
Green House-Geisser correction
Which post hoc test is most robust and most conservative for a repeated measures ANOVA?
Dunnets
If parametric assumptions are in doubt, we must use the non-parametric equivalent of a single factor repeated measures ANOVA which is:
Durbin-Watson test
Friedman test
Hochberg's GT2 test
For which analysis do BOTH the equality of variance assumption and sphericity assumption apply?
Non-linear regression
Two-way ANOVA
Independent ANOVA
Mixed model ANOVA
Repeated measures ANOVA
In terms of polynomial regression, what happens if you add further terms to the polynomial? (multiple correct answers)
the fit will automatically improve
there is a risk of over-fitting the model
the significance level associated with the F-statistic changes
the R-squared value will increase
In terms of nonlinear regression, why would you want to try multiple starting parameters?
to ensure that the interaction between the variables is taken into account
to ensure that the computer has found the global minimum
to ensure that the computer has found the local minimum
to ensure that an accurate scientific relationship is found
How would you calculate the Sum of Squares (SS)?
add all the standard deviations together and square that value
square the mean from each sample and add those together
square each standard deviation and add them all together
square each standard deviation and add this to the variance
Variance is calculated by doing what?
dividing the standard deviations by the degrees of freedom
dividing the sum of squares by the degrees of freedom
multiplying the degrees of freedom by the mean
multiplying the standard deviations by the sum of squares
How do we define the normal distribution curve?
the population mean is the height and the sum of squares is the distance from the midline of the curve to the edge
the variance is the height and the population mean is the distance from the midline of the curve to the edge
the population standard deviation is the height and the population mean is the distance from the midline of the curve to the edge
the population mean is the height and the population standard deviation is the distance from the midline of the curve to the edge
How do you calculate a z-score?
(x - mean) /sd
(x - sd)/mean
(mean-x)/sd
(x + mean)/sd
Choose all of the correct statements
approximately 99% of normally-distributed values lie between +- 2 sds from the mean
approximately 95% of normally-distributed values lie between +-2 sds from the mean
approximately 99.9% of normally-distributed values lie between +- 2.6 sds from the mean
approximately 99% of normally-distributed values lie between +- 2.6 sds from the mean
approximately 99.9% of normally-distributed values lie between +- 3 sds from the mean
approximately 95% of normally-distributed values lie between +- 3 sds from the mean
How do you calculate SEM and therefore, confidence intervals?
SEM = sd x square root of n and therefore a 95% CI would be +- 1.96 x SEM
SEM = sd/square root of n and therefore a 95% CI would be +- 3 x SEM
SEM = sd x square root of n and therefore a 95% CI would be +- 2.6 x SEM
SEM = sd/square root of n and therefore a 95% CI would be +- 1.96 x SEM
Which statement is true?
P < 0.05 means that 5% of the results arose by chance if the null hypothesis is true
P < 0.05 means <5% probability of the results arising by chance if the null hypothesis is true
P < 0.05 means <0.05% probability of the results arising by chance if the null hypothesis is true
P < 0.05 means that <0.5% probability of the results arising by chance if the null hypothesis is true
Choose the correct statements
type 1 error rate is conventionally set to 5% ( P < 0.05)
type 2 error rate is conventionally set to 5% ( P < 0.05)
type 1 error rate = 1 - power
type 2 error rate = 1 - power
if you accept a statistical power of 80% it will mean that you have a type 2 error rate of 20%
if you accept a statistical power of 80% it will mean that you have a type 1 error rate of 20%
What happens if you design an experiment with 3 groups and are tempted to test for differences between the means using 3 separate t-tests? (multiple correct answers)
you will increase the chance of making a type 2 error
you will increase the chance of making a type 1 error
you will inflate your p-value
you will decrease your p-value
Which statements are correct regarding the Pearson Correlation Coefficient?
+- 0.5 is a large effect
+- 0.1 is a small effect
+- 1 is a small effect
it measures how close the data points are to a straight line that best describes the linear relationship
r = +0.1 refers to a perfect straight line with a positive slope
r = -1 refers to a perfect straight line with a negative slope
How is the line of best fit created in simple linear regression?
by minimising the total sum of squares
by minimising the sum of squares of the residuals
by creating an equation which fits the model best
by entering the data into the computer in Hierarchical form
How do you calculate R squared?
1 - (SS of the residuals/total SS)
1 + (SS of the residuals/total SS)
1 - (total SS/SS of the residuals)
1 + (total SS/SS of the residuals)
Shapiro Wilk's test is used to...
check for sphericity
correct degrees of freedom
ascertain that residuals are random and normally distributed
minimise the sum of squares of the residuals
When entering more than 2 categories as dummy variables... (multiple correct answers)
the thing that you're comparing the baseline to gets a 1
the thing that you're comparing the baseline to gets a 0.1
1 fewer dummy variables than number of categories
baseline condition gets a value of 0
baseline condition gets a value of 1.5
Bonferroni test on its own - the p-values need to be less than what to claim significance?
0.05/number of categories
0.05/n
0.05/number of comparisons
0.05/variance
Normally distributed variables X and Y are significantly correlated with a p level of 0.006 and a Pearson’s correlation coefficient of 0.468. Approximately how much of the variability in X and Y can be explained by this correlation?
32%
47%
22%
13%
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