outcome (continuous variable) was natural log-transformed, back-transform result (β) when interpreting results?

[Hui SHI]

Dear experts,

I express my sincere thank for this forum and helpful experts in this forum.

Recently, I fit a multiple linear regression model with depression (measured by CESD score) as outcome. Since this study was conducted among young women, the depression showed a right-skewed distribution. So when I fit the linear model and check the residual, I found that this model violated the normality of residuals assumption in linear regression. So depression was natural log-transformed (Logdepres) and fit the linear regression again using Logdepres. This time, the graph of residuals looks greater than no-transformed model.

My question is: In linear regression model,when outcome (continuous variable) was natural log-transformed, do I need to back-transform result (β) when interpreting results?

I attached part of my results. I plan to back-transform each β as: exp(β).

Some professor suggested to do back-transform, others think the values of back-transformed are not true. I have reviewed some articles as followed which also mentioned back-transformed for interpreting.

1. Association of Sleep Characteristics and Cognition in Older Community-Dwelling Men: the MrOS Sleep Study
2. Relationships Between Sleep Stages and Changes in Cognitive Function in Older Men: The MrOS Sleep Study
3. Associations Between Sleep-Disordered Breathing, Nocturnal Hypoxemia, and Subsequent Cognitive Decline in Older Community-Dwelling Men: The Osteoporotic Fractures in Men Sleep Study
4. Nonparametric Parameters of 24-Hour Rest–Activity Rhythms and Long-Term Cognitive Decline and Incident Cognitive Impairment in Older Men
5. Associations of 24-Hour Light Exposure and Activity Patterns and Risk of Cognitive Impairment and Decline in Older Men: The MrOS Sleep Study
6. Associations of Sleep Architecture and Sleep Disordered Breathing with Cognition in Older Community-Dwelling Men: The MrOS Sleep Study

In these 6 paper, authors described as followed in their Statistical Analysis sections.

The continuous cognitive scores were transformed to meet model requirements (log transformation for Trails B, cube
transformation for 3MS) and back-transformed for display of results.
or
Log transformation was performed on Trails B scores to improve the normality of the distribution, and the results were
back-transformed to the original scale.

So I also planned to do back-transformation of β. But I just feel a little weird for back-transformed results.

Thanks very much.

I am looking forward to your constructive opinion!

[Rich Goldstein]

the coefficients here tell you about differences in the logged variable - is that really what you want; you will probably be better off using -poisson-; see the following blog: https://blog.stata.com/2011/08/22/us...tell-a-friend/

[Hui SHI]

the coefficients here tell you about differences in the logged variable - is that really what you want; you will probably be better off using -poisson-; see the following blog: https://blog.stata.com/2011/08/22/us...tell-a-friend/

Hi Rich,

Thanks for your quick response, and thanks for your shared blog!

What would you suggest I do if I must use linear regression to fit the model in some condition.

Because I remember I have same question when I fit isotemporal substitution framework (ISM) and compositional data analysis (CoDA), which two approach included linear regression.


Thanks again!

[Hui SHI]

Dear experts,

I express my sincere thank for this forum and helpful experts in this forum.

Recently, I fit a multiple linear regression model with depression (measured by CESD score) as outcome. Since this study was conducted among young women, the depression showed a right-skewed distribution. So when I fit the linear model and check the residual, I found that this model violated the normality of residuals assumption in linear regression. So depression was natural log-transformed (Logdepres) and fit the linear regression again using Logdepres. This time, the graph of residuals looks greater than no-transformed model.

My question is: In linear regression model,when outcome (continuous variable) was natural log-transformed, do I need to back-transform result (β) when interpreting results?

I attached part of my results. I plan to back-transform each β as: exp(β).

Some professor suggested to do back-transform, others think the values of back-transformed are not true. I have reviewed some articles as followed which also mentioned back-transformed for interpreting.

1. Association of Sleep Characteristics and Cognition in Older Community-Dwelling Men: the MrOS Sleep Study
2. Relationships Between Sleep Stages and Changes in Cognitive Function in Older Men: The MrOS Sleep Study
3. Associations Between Sleep-Disordered Breathing, Nocturnal Hypoxemia, and Subsequent Cognitive Decline in Older Community-Dwelling Men: The Osteoporotic Fractures in Men Sleep Study
4. Nonparametric Parameters of 24-Hour Rest–Activity Rhythms and Long-Term Cognitive Decline and Incident Cognitive Impairment in Older Men
5. Associations of 24-Hour Light Exposure and Activity Patterns and Risk of Cognitive Impairment and Decline in Older Men: The MrOS Sleep Study
6. Associations of Sleep Architecture and Sleep Disordered Breathing with Cognition in Older Community-Dwelling Men: The MrOS Sleep Study

In these 6 paper, authors described as followed in their Statistical Analysis sections.

The continuous cognitive scores were transformed to meet model requirements (log transformation for Trails B, cube
transformation for 3MS) and back-transformed for display of results.
or
Log transformation was performed on Trails B scores to improve the normality of the distribution, and the results were
back-transformed to the original scale.

So I also planned to do back-transformation of β. But I just feel a little weird for back-transformed results.

Thanks very much.

I am looking forward to your constructive opinion!

I have an another question: if the normality of residuals assumption was violated due to predictor/covariates, we did the same natural log-transformed, can we back-transform result (β) when interpreting results?