Study Design: From group comparisons to individual predictions

Question 1

What are the 2 defining properties of Gaussian Distribution?

Answer 1

1. The middle of the distribution mirrored from left to right - denoted by mew
2. The standard deviation which defines essentially the width of the curve and uses the Greek letter sigma

Question 2

What is statistical significance?

Answer 2

1. The Gaussian distribution has tails which go to +/- infinity - there is a finite probability of extremely unlikely events (although the statistical models may break down at the extremes)
2. With the Gaussian distribution no result is impossible. Instead, we set a limit of statistical significance, usually at p=0.05
3. For a Gaussian distribution, this occurs at z=1.96 I.e. if a result is more than 2 standard deviation from the mean, it’s likely to be significantly different from the mean
   - we observe values that are extreme
   - For a normal standard distribution, we talk about z score - mean zero and standard deviation 1

Question 3

What is the Null Hypothesis?

Answer 3

1. Compute a probability of everything on the left side
2. The p-value is the area under the whole curve
3. The 5% cut-off = 1.96
4. To rest for statistical significance, we set up a Null hypothesis H0. This is the opposite to the hypothesis that we wish to test
5. If the value is 2 standard deviation away from the mean of distribution, then we can call it statistically significant - essentially p value is too small
6. The p-value is the probability of obtaining the observed result (or greater) If the null hypothesis is true

Question 4

What does the P-value >0.05 and <0.05 mean?

Answer 4

1. If P-value > 0.05, the probability of obtaining this result is greater than 1/20 or 5% so we assume cannot reject the Null hypothesis
2. If the P-value < 0.05, the probability of obtaining this result is < 1/20 or % so we say that the result is unlikely to have obtained if the Null hypothesis is true, so we can reject the Null hypothesis

Question 5

What is an arbitrary threshold with no real meaning?

Answer 5

5%

Question 6

What can be very misleading?

Answer 6

P-values and the significance test

Question 7

What is common to ask in medical statistics?

Answer 7

Whether two groups are different

E.g. we might try a new treatment for headache on a group of people and compare the outcome to a group who received the placebo. We want to know if the groups are significantly different (the treatment may make the headache worse)

Question 8

What is the most commonly used test?

Answer 8

Students t-test

Question 9

What is Student’s t-test based on?

Answer 9

Student’s t-distribution:

• “heavier tail” than the the normal distribution
• has one parameter: df (degrees of freedom)
• mean 0

Question 10

What is students t-test I?

Answer 10

Students t-test is commonly used to compare two samples, with me and standard deviation and number of measurements respectively. Then we need an estimate of the combined standard error of mean

Question 11

What is students t-test II?

Answer 11

Strictly, both distribution should be Gaussian and have the same standard deviation, but student t-test is robust and works well for non-Gaussian distributions

Note: Welch’s test is more general (and implemented in more statistics software)

Question 12

What is Paired t-test?

Answer 12

Used with matched samples, e.g. participants before and after treatment
- used to compare the means

Question 13

What is longitudinal data?

Answer 13

Observing people over time

Conceptually, longitudinal imaging data allows to conduct ‘paired tests’

Gain in statistical power

• uses the baseline data as a reference for each subject
• less susceptible to noise between subjects
• can be combined with a reference group or intervention etc

Question 14

What is longitudinal imaging data?

Answer 14

1. Co-registration and substation of longitudinal volumetric scans
2. Serial hippocampal volumetric
   - boundary shift integral - looking at defining boundaries in terms of the hippocampus
   - manually hand traced

Question 15

What are the problems with longitudinal imaging data?

Answer 15

1. Takes a long time to acquire the data
2. The scanners can break
3. The study is longer

Question 16

What is individual predictions?

Answer 16

1. Classical statistical tests are often comparisons between groups (e.g. patients vs controls)
2. Patients with Alzheimers have on average reduced hippocampal volume by 22%
3. In order to provide individual predictions, one must first learn a model, which can be used to make a personalised prediction
4. Essentially this is machine learning
5. If we formalise this:
   We have an input: X (feature matrix)
   And target Y - continuous (regression), discrete groups (classification)
6. In predictive settings:
   Aim to predict Y from the information given in C

Question 17

What are the features of imaging-settings?

Answer 17

1. X represents the voxel intensities
2. X summarises of ROIs
3. Y represents disease groups (HC, MCI, AD)
4. Y represents a cognitive trait) e.g. MMSE

Question 18

What is the job in predictive modelling?

Answer 18

Our job is to find these Bis
We use training data for this and find the best combination of Bi so that our model f(x) is close to the observation y
Equipped with f(x) we can now make predictions

Question 19

What does the Linear regression closely related to?

Answer 19

1. ‘Classic’ statistics
2. Each Bi tells us how strongly an input influence the output
3. One can compute how reliable the Bi were estimated
4. This can be turned into a Z-score and a p-value

Question 20

What is shrinkage methods?

Answer 20

1. Shrinkage methods: shrink coefficient Bj
2. Achieved by imposing a penalty
3. Ridge regression

Question 21

What is Lasso regression?

Answer 21

1. Regression analysis method that performs both variable selection and regularisation in order to enhance the prediction accuracy and interpret ability of the statistical model it produces
2. Lasso was originally formulated for least squares models and this simple case reveals a substantial amount about the behaviour of the estimator, including its relationship to ridge regression and best subset selection and the connections between lasso coefficient estimates and so called soft threshold of
3. It also reveals that the coefficient estimates do not need to be unique if the co-variables are collinear

Question 22

What is pairwise scatter plot?

Answer 22

Matrix of scatter plots similar to a correlation matrix

Each pair of numeric variable is displayed vs the other one. It can quickly help us establish relationships

Question 23

What is ordinary least squares (OLS)?

Answer 23

Is a type of linear least squares method for estimating the unknown parameters in a linear regression model

OLS chooses the parameters of a linear function of a set of explanatory variables by the principles of least squares: minimising the sum of the squares of the differences between the observed dependent variable in the given dataset and those predicted by linear function

Question 24

What is sparse model?

Answer 24

Certain features are eradicated
Setting up specific beta-values exactly to 0
This is where the lasso estimate comes in