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6. Consider a clinical trial evaluating a new talking therapy intervention, del...

Jun 5, 2024

Solution by Steps

step 1

Define the outcome variable for the

i

-th participant in arm

k

of the trial as

y_{ik}

step 2

Let

x_{ik}

be an indicator variable that equals 1 if participant

i

is in the intervention arm and 0 otherwise

step 3

Let

u_j

be the random effect for therapist

j

, where

j

ranges from 1 to 6

step 4

The mixed model for

y_{ik}

can be written as:

y_{ik} = \beta_0 + \beta_1 x_{ik} + \sum_{j=1}^6 u_j d_{ij} + \epsilon_{ik}

where

d_{ij}

is the number of sessions delivered by therapist

j

to participant

i

, and

\epsilon_{ik}

is the residual error term

step 5

Assumptions: 1. The random effects

u_j

are normally distributed with mean 0 and variance

\sigma_u^2

. 2. The residual errors

\epsilon_{ik}

are normally distributed with mean 0 and variance

\sigma_\epsilon^2

. 3. The random effects

u_j

and residual errors

\epsilon_{ik}

are independent

Part (b)

step 1

The variance of

y_{ik}

in the intervention arm can be derived as follows:

\text{Var}(y_{ik}) = \text{Var}(\beta_0 + \beta_1 x_{ik} + \sum_{j=1}^6 u_j d_{ij} + \epsilon_{ik})

Since

\beta_0

and

\beta_1 x_{ik}

are fixed effects, their variances are 0

step 2

Therefore,

\text{Var}(y_{ik}) = \text{Var}(\sum_{j=1}^6 u_j d_{ij} + \epsilon_{ik})

= \sum_{j=1}^6 d_{ij}^2 \sigma_u^2 + \sigma_\epsilon^2

step 3

The correlation between

y_{ik}

and

y_{i'k}

in the intervention arm is given by:

\text{Corr}(y_{ik}, y_{i'k}) = \frac{\text{Cov}(y_{ik}, y_{i'k})}{\sqrt{\text{Var}(y_{ik}) \text{Var}(y_{i'k})}}

\text{Cov}(y_{ik}, y_{i'k}) = \text{Cov}(\sum_{j=1}^6 u_j d_{ij} + \epsilon_{ik}, \sum_{j=1}^6 u_j d_{i'j} + \epsilon_{i'k})

Since

\epsilon_{ik}

and

\epsilon_{i'k}

are independent, their covariance is 0

step 4

Therefore,

\text{Cov}(y_{ik}, y_{i'k}) = \sum_{j=1}^6 d_{ij} d_{i'j} \sigma_u^2

\text{Corr}(y_{ik}, y_{i'k}) = \frac{\sum_{j=1}^6 d_{ij} d_{i'j} \sigma_u^2}{\sqrt{(\sum_{j=1}^6 d_{ij}^2 \sigma_u^2 + \sigma_\epsilon^2)(\sum_{j=1}^6 d_{i'j}^2 \sigma_u^2 + \sigma_\epsilon^2)}}

Part (c)

step 1

Given the between-therapist variance

\sigma_u^2 = 2

, we need to determine the covariance between patients 2 and 3

step 2

From the data, the therapists delivering sessions to patients 2 and 3 are: - Patient 2: Therapists 1, 5, 4 - Patient 3: (data not fully provided, assume similar structure)

step 3

Assuming the therapists for patient 3 are also 1, 5, 4, the covariance is:

\text{Cov}(y_{2k}, y_{3k}) = \sum_{j=1}^6 d_{2j} d_{3j} \sigma_u^2

= (1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1) \cdot 2 = 3 \cdot 2 = 6

Part (d)

step 1

The formula component of the brms code to estimate the treatment effect can be written as:

\text{bf}(y \sim trt + (1 | therapist))

where `trt` is the treatment indicator and `therapist` is the random effect for therapists

step 2

A criterion to decide if the treatment should be adopted in practice could be the statistical significance of the treatment effect (e.g., p-value < 0.05) and the clinical significance (e.g., effect size)

Part (e)

step 1

To minimize the variance of a patient's outcome, the allocation scheme should ensure that the same therapist delivers all treatment sessions for that patient

step 2

This minimizes the variance because it eliminates the between-therapist variance component, leaving only the residual error variance

Answer

The mixed model for the outcome variable, variance and correlation expressions, covariance calculation, brms code, and optimal allocation scheme are provided.

Key Concept

Mixed Model and Variance Calculation

Explanation

The mixed model accounts for both fixed and random effects, and the variance and correlation expressions help understand the impact of clustering due to therapists.