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You should note that T is none of your model's a random effects terms, but a fixed effect. Random effects are only those effects that appear after the | in a lmer formula!

A more thorough discussion of what this specification does you can find in this lmer faq question.

From this questions your model should give the following (for your fixed effect T):

• A global slope
• A random slopes term specifying the deviation from the overall slope for each level of Site
• The correlation between the random slopes.

And as said by @mark999 this indeed is a common specification. In repeated measures designs, you generally want to have random slopes and correlations for all repeated measures (within-subjects) factors.

See the following paper for some examples (which I tend to always cite here):

Judd, C. M., Westfall, J., & Kenny, D. A. (2012). Treating stimuli as a random factor in social psychology: A new and comprehensive solution to a pervasive but largely ignored problem. Journal of Personality and Social Psychology, 103(1), 54–69. doi:10.1037/a0028347

You should note that T is none of your model's a random effects terms, but a fixed effect. Random effects are only those effects that appear after the | in a lmer formula!

A more thorough discussion of what this specification does you can find in this lmer faq question.

From this questions your model should give the following (for your fixed effect T):

• A global slope
• A random slopes term specifying the deviation from the overall slope for each level of Site
• The correlation between the random slopes.

And as said by @mark999 this indeed is a common specification. In repeated measures designs, you generally want to have random slopes and correlations for all repeated measures (within-subjects) factors.

See the following paper for some examples (which I tend to always cite here):

Judd, C. M., Westfall, J., & Kenny, D. A. (2012). Treating stimuli as a random factor in social psychology: A new and comprehensive solution to a pervasive but largely ignored problem. Journal of Personality and Social Psychology, 103(1), 54–69. doi:10.1037/a0028347

You should note that T is in none of your model's a random effects terms, but a fixed effect. Random effects are only those effects that appear after the | in a lmer formula!

A more thorough discussion of what this specification does you can find in this lmer faq question.

From this questions your model should give the following (for your fixed effect T):

• A global slope
• A random slopes term specifying the deviation from the overall slope for each level of T
• The correlation between the random slopes.

And as said by @mark999 this indeed is a common specification. In repeated measures designs, you generally want to have random slopes and correlations for all repeated measures (within-subjects) factors.

See the following paper for some examples (which I tend to always cite here):

Judd, C. M., Westfall, J., & Kenny, D. A. (2012). Treating stimuli as a random factor in social psychology: A new and comprehensive solution to a pervasive but largely ignored problem. Journal of Personality and Social Psychology, 103(1), 54–69. doi:10.1037/a0028347

You should note that T is none of your model's a random effects terms, but a fixed effect. Random effects are only those effects that appear after the | in a lmer formula!

A more thorough discussion of what this specification does you can find in this lmer faq question.

From this questions your model should give the following (for your fixed effect T):

• A global slope
• A random slopes term specifying the deviation from the overall slope for each level of Site
• The correlation between the random slopes.

And as said by @mark999 this indeed is a common specification. In repeated measures designs, you generally want to have random slopes and correlations for all repeated measures (within-subjects) factors.

See the following paper for some examples (which I tend to always cite here):

Judd, C. M., Westfall, J., & Kenny, D. A. (2012). Treating stimuli as a random factor in social psychology: A new and comprehensive solution to a pervasive but largely ignored problem. Journal of Personality and Social Psychology, 103(1), 54–69. doi:10.1037/a0028347

You should note that T is in none of your models a random effects term, but a fixed effects term. Random effects are only those effects that appear after the | in a lmer formula.

A more thorough discussion of what this specification does you can find in this lmer faq question.

From this questions your model should give the following (for T):

• A global slope
• A random slopes term specifying the deviation from the overall slope for each level of T
• The correlation between the random slopes.

And as said by @mark999 this indeed is a common specification. In repeated measures designs, you generally want to have random slopes and correlations for all repeated measures (within-subjects) factors.

See the following paper for some examples (which I tend to always cite here):

Judd, C. M., Westfall, J., & Kenny, D. A. (2012). Treating stimuli as a random factor in social psychology: A new and comprehensive solution to a pervasive but largely ignored problem. Journal of Personality and Social Psychology, 103(1), 54–69. doi:10.1037/a0028347

You should note that T is in none of your model's a random effects terms, but a fixed effect. Random effects are only those effects that appear after the | in a lmer formula!

A more thorough discussion of what this specification does you can find in this lmer faq question.

From this questions your model should give the following (for your fixed effect T):

• A global slope
• A random slopes term specifying the deviation from the overall slope for each level of T
• The correlation between the random slopes.

And as said by @mark999 this indeed is a common specification. In repeated measures designs, you generally want to have random slopes and correlations for all repeated measures (within-subjects) factors.

See the following paper for some examples (which I tend to always cite here):

Judd, C. M., Westfall, J., & Kenny, D. A. (2012). Treating stimuli as a random factor in social psychology: A new and comprehensive solution to a pervasive but largely ignored problem. Journal of Personality and Social Psychology, 103(1), 54–69. doi:10.1037/a0028347