DF in lme

Ben

Suppose you calculated an average response for each of the 30 plates in
your experiment, and calculated a standard two-way anova as follows:

Source of variation    DF
Groups            1
Lines            4
Groups x lines        4
Residual        20

The F-tests from this anova should agree with the Wald tests from lmer.
The residual is based on variation between plates within lines and
groups. If I understand the design correctly, the other sources of
variation (between disks in plates, between  readings within disks) may
be of interest but do not feature individually in the testing of groups
and lines.

When data are balanced, an anova can clarify some of the obscurities of
mixed model fitting. Is this a controversial observation on this list?

Ben Ward wrote:

On 16/03/2011 15:08, Ben Bolker wrote:

On 11-03-16 03:52 AM, Ben Ward wrote:

Hi, I'm using lme and lmer in my dissertation and it's the first time
I've used these methods. Taking into account replies from my previous
query I decided to go through with a model simplification, and then try
to validate the models in various ways and come up with the best one to
include in my work, be it a linear mixed effects model or general
linear
effects model, with log() data or not etc - interestingly it does not
seems like doing transofrmations and such makes much difference so far,
looking at changes in diagnostic plots and AIC.

   Be careful about comparing fits of transformed and non-transformed
data via AIC/log-likelihood: e.g. see
<http://www.unc.edu/courses/2006spring/ecol/145/001/docs/lectures/lecture18.htm>.

  (This does *not* refer to the link function, e.g. the log link of the
Poisson, but to the case where you transform your data prior to
analysis.)

Anywho, I simplified to the model using lme (I've pasted it at the
bottom). And looking at the anova output the numDF looks right. However
I'm concerned about the 342 df in the denDF in anova() and in the
summary() output, as it seems to high to me, because at the observation
level is too high and pseudoreplicated; 4 readings per disk, 3 disks,
per plate, 3 plates per lineage, 5 lineages per group, 2 groups so:
4*3*3*5*2=360. If I take this to disk level 3*3*5*2=90, and at dish
level it's 3*5*2=30 degrees of freedom for error. And either dish or
disk (arguments for both) is the level at which one independant
point of
datum is obtained, most probably Dish. So I'm wondering if either I'de
done something wrong, or I'm not understanding how df are presented and
used in mixed models. It's not really explained in my texts, and my
lecturer told me I'm working at the edge of his personal/professional
experience.

    At what level are Group and Lineage replicated in the model?  Do you
have different Groups or Lineages represented on the same disk, dish, or
plate?  If you do have multiple Groups and Lineages present at the
lowest level of replication, then you have a randomized block design and
the degrees of freedom may be higher than you think.  If you really want
denominator degrees of freedom and you want them correct, consult an
experimental design book and figure out what they should be in the
classical framework ...

I'm now of the opinion that - (Just trying to get my head around it) -
that I don't have a randomized block design:
I've done a bit like a lenski evolution experiment with my microbes,
which involed two groups, in those two groups i have 5 cultures each,
one group is 5 lineages of bacteria I have been evolving against some
antimicrobial, the other group have not been through this - they are
stock run of the mill organisms. So with those 5 cultures of evolved
bacteria, for each, I'd take some, and spread it on three plates - so
theres no intermingling or randomization/mixing of the cultures: each
gets plated onto a who plate itself three times. Then the three disks,
loaded with antimicrobial were loaded onto each plate, and they were
incubated, and then I took 4 measurements from each zone that formed
around those disks. The disks all have the same antimicrobial on them.
So in that way, if what you say by randomized block design is
something like a split plot experiment, where there are several plots,
and numerous plants, and each one got a different treatment, then I
don't believe my experiment is like that.  In my case that would be
like me having different cultures on the same dish, or using disks
with different antimicrobials on, at least I think this is what you're
asking. In which case Dish is the level at which I get truly
indepentent pieces of data, and 3plates*5lineages*2Groups=30: If I
recode my factor levels then, like so, which I mentioned before as a
possibility:
Diameter<-Dataset$Diameter
Group<-factor(Dataset$Group)
Lineage<-factor(Dataset$Lineage)
Dish<-factor(Dataset$Dish)
Disk<-factor(Dataset$Disk)
lineage<-Group:Lineage
dish<-Group:Lineage:Dish
disk<-Group:Lineage:Dish:Disk

 And then fit the model:

model <- lme(Diameter~Group*Lineage,random=~1|dish/disk, method="REML")

I get the following:

 > summary(model)

Linear mixed-effects model fit by REML
 Data: NULL
       AIC      BIC    logLik
  1144.193 1194.346 -559.0966

Random effects:
 Formula: ~1 | dish
        (Intercept)
StdDev:   0.2334716

 Formula: ~1 | disk %in% dish
        (Intercept) Residual
StdDev:    0.356117 1.079568

Fixed effects: Diameter ~ Group * Lineage
                                        Value Std.Error  DF  t-value
p-value
(Intercept)                         15.049722 0.2542337 270 59.19641 
0.0000
Group[T.NEDettol]                    0.980556 0.3595407  20  2.72724 
0.0130
Lineage[T.First]                    -0.116389 0.3595407  20 -0.32372 
0.7495
Lineage[T.Fourth]                   -0.038056 0.3595407  20 -0.10584 
0.9168
Lineage[T.Second]                   -0.177500 0.3595407  20 -0.49369 
0.6269
Lineage[T.Third]                     0.221111 0.3595407  20  0.61498 
0.5455
Group[T.NEDettol]:Lineage[T.First]   2.275000 0.5084674  20  4.47423 
0.0002
Group[T.NEDettol]:Lineage[T.Fourth]  0.955556 0.5084674  20  1.87929 
0.0749
Group[T.NEDettol]:Lineage[T.Second]  0.828333 0.5084674  20  1.62908 
0.1189
Group[T.NEDettol]:Lineage[T.Third]   0.721667 0.5084674  20  1.41930 
0.1712
 Correlation:
                                    (Intr) Gr[T.NED] Lng[T.Frs]
Lng[T.Frt]
Group[T.NEDettol]                   -0.707
Lineage[T.First]                    -0.707  0.500
Lineage[T.Fourth]                   -0.707  0.500     0.500
Lineage[T.Second]                   -0.707  0.500     0.500      0.500
Lineage[T.Third]                    -0.707  0.500     0.500      0.500
Group[T.NEDettol]:Lineage[T.First]   0.500 -0.707    -0.707     -0.354
Group[T.NEDettol]:Lineage[T.Fourth]  0.500 -0.707    -0.354     -0.707
Group[T.NEDettol]:Lineage[T.Second]  0.500 -0.707    -0.354     -0.354
Group[T.NEDettol]:Lineage[T.Third]   0.500 -0.707    -0.354     -0.354
                                    L[T.S] L[T.T]
Grp[T.NEDttl]:Lng[T.Frs]
Group[T.NEDettol]
Lineage[T.First]
Lineage[T.Fourth]
Lineage[T.Second]
Lineage[T.Third]                     0.500
Group[T.NEDettol]:Lineage[T.First]  -0.354 -0.354
Group[T.NEDettol]:Lineage[T.Fourth] -0.354 -0.354  0.500
Group[T.NEDettol]:Lineage[T.Second] -0.707 -0.354  0.500
Group[T.NEDettol]:Lineage[T.Third]  -0.354 -0.707  0.500
                                    Grp[T.NEDttl]:Lng[T.Frt]
G[T.NED]:L[T.S
Group[T.NEDettol]
Lineage[T.First]
Lineage[T.Fourth]
Lineage[T.Second]
Lineage[T.Third]
Group[T.NEDettol]:Lineage[T.First]
Group[T.NEDettol]:Lineage[T.Fourth]
Group[T.NEDettol]:Lineage[T.Second]  0.500
Group[T.NEDettol]:Lineage[T.Third]   0.500                    0.500

Standardized Within-Group Residuals:
        Min          Q1         Med          Q3         Max
-2.26060119 -0.70948250  0.03630884  0.69899536  3.42475990

Number of Observations: 360
Number of Groups:
          dish disk %in% dish
            30             90

 > anova(model)

              numDF denDF  F-value p-value
(Intercept)       1   270 39586.82 <.0001
Group             1    20   145.07 <.0001
Lineage           4    20     4.58  0.0087
Group:Lineage     4    20     5.27  0.0046

This is closer to what I was expecting in terms of DF: 3 plates*5
lineages=15: 15 samples per group, 15-4(the numDF Lineage)=11,
11-1(the numDF for Group)= 10 x 2 for the two groups/treatments = 20.
Hopefully I've worked that out correctly, and sombody could tell me
whether . Its' awkward because this experiment is unprecedented at my
uni, it was offered up by a teacher as a topic but then got dropped
due to lack of interest. As it's the first time, myself and my
supervisor were in many ways flying blind. If I remove the Lineage
main effect term, and include it as a random effect, leaving only
group as a fixed effect:

 > anova(model2)

            numDF denDF  F-value p-value
(Intercept)     1   270 8041.429 <.0001
Group           1     8   29.469   6e-04

I get 8DF which by the same reasoning in the above model, is 5-1=4,
4*2 = 8, so I take that as reassurance my working is correct. I'd also
like to ask for opinion, on whether it would be advisable to actually
remove lineage as a fixed effect, and include lineage as a random
effect on the slope, rather than intersect which is what I've put all
the others as. I ask this because, whilst I feel whilst lineage might
seem a factor with informative levels( tha's how I first saw them), I
had no way of predicting which ones would show greatest or smallest
differences or how the five factor levels would interact and shape my
data, in that way the factor levels are not really all that
informative at all - they're just numbered as dish and disk are, and
their effects may even be different within my two groups - they don't
really allow any prediction in the same way a factor for different
types of fertiliser would in a plant study would for example, so I'm
thinking maybe it should be a random effect.

Thank you very much to everyone that's replied to me and assisted me
with this, it's a tough learning curve, but I do think I'm beginning
to grasp how to use lme and lmer for my basic ends. Once I'm confident
on the above, I'm next considering, whether to try an introduce some
weighting options to see what happens to a small amount of
heterscedacity I have between the two groups.

Ben W.

I've used lmer and the function in languageR to extract p-values
without
it even mentioning df. Now if the lmer method with pvals.fnc() makes it
so as I don't have to worry about these df then in a way it makes my
issue a bit redundant. But it is playing on my mind a bit so felt I
should ask.

My second question is about when I do the equivalent model using lmer:
"lmer(Diameter~Group*Lineage+(1|Dish)+(1|Disk), data=Dataset)" - which
I'm sure does the same because all my plots of residuals against fitted
and such are the same, if I define it with the poisson family, which
uses log, then I get a much lower AIC of about 45, compared to over
1000
without family defined, which I think defaults to gaussian/normal.

   I don't think you should try to pick the family on the basis of
AIC --
you should pick it on the basis of the qualitative nature of the data.
If you have count data, you should probably use Poisson (but you may
want to add an observation-level random effect to allow for
overdispersion.)  If your response variable is Diameter, it is **not** a
count variable, and you shouldn't use Poisson -- you should use an
appropriately transformed response variable.

I've tried transforming my response variable in a few ways, like
natural log, sqrt, and (x/1) but they don't really seem to alter the
distribution or shape of my data at all.
Interestingly, if I look at the spread of the data by splitting the
response variable between the two groups, I see much more symmetry -
although still not a nice neat normal distribution, but in Biology
I've been taught never to expect one.

  And

my diagnostic plots still give me all the same patters, but just
looking
a bit different because of the family distribution specified. I then
did
a model logging the response variable by using log(Diameter), again, I
get the same diagnostic plot patterns, but on a different scale, and I
get an AIC of - 795.6. Now normally I'd go for the model with the
lowest
AIC, however, I've never observed this beahviour before, and can't help
but think thhat the shift from a posotive 1000+ AIC to a negative
one is
due to the fact the data has been logged, rather than that the model
fitted to log data in this way is genuinley better.

Finally, I saw in a text, an example of using lmer but "Recoding Factor
Levels" like:
lineage<-Group:Lineage
dish<-Group:Lineage:Dish
disk<-Group:Lineage:Dish:Disk
model<-lmer(Diameter~Group+(1|lineage)+(1|dish)+(1|disk)

However I don't see why this should need to be done, considering, the
study was hieracheal, just like all other examples in that chapter, and
it does not give a reason why, but says it does the same job as a
nested
anova, which I though mixed models did anyway.

   (1|lineage)+(1|dish)+(1|disk)

   is the same as

   (1|Lineage/Dish/Disk)

   (1|Dish) + (1|Disk) is **not** the same as (1|Dish/Disk), if Disk is
not labeled uniquely (i.e. if Dishes are A, B, C, .. and Disks are 1, 2,
3, ... then you need Dish/Disk.  If you have labeled Disks A1, A2, ...
B1, B2, ... then the specifications are equivalent.

   For a linear mixed model (i.e. not Poisson counts) you should be able
to run the same model in lmer and lme and get extremely similar results.

Hopefully sombody can shed light on my concerns. In terms of my work
and
university, I could include what I've done here and be as transparrant
as possible and discuss these issues, because log() of the data or
defining a distribution in the model is leading to the same plots and
conclusions. But I'd like to make sure I come to term with what's
actually happening here.

A million thanks,
Ben W.


lme14<- lme(Diameter~Group*Lineage,random=~1|Dish/Disk, data=Dataset,
method="REML")

anova(lme14):

  numDF denDF   F-value p-value
(Intercept)       1   342 16538.253<.0001
Group             1   342   260.793<.0001
Lineage           4   342     8.226<.0001
Group:Lineage     4   342     9.473<.0001

summary(lme14)

Linear mixed-effects model fit by REML
  Data: Dataset
        AIC      BIC    logLik
   1148.317 1198.470 -561.1587

Random effects:
  Formula: ~1 | Dish
         (Intercept)
StdDev:   0.1887527

  Formula: ~1 | Disk %in% Dish
          (Intercept) Residual
StdDev: 6.303059e-05 1.137701

Fixed effects: Diameter ~ Group * Lineage
                                         Value Std.Error  DF  t-value
p-value
(Intercept)                         15.049722 0.2187016 342 68.81396
0.0000
Group[T.NEDettol]                    0.980556 0.2681586 342  3.65662
0.0003
Lineage[T.First]                    -0.116389 0.2681586 342 -0.43403
0.6645
Lineage[T.Fourth]                   -0.038056 0.2681586 342 -0.14191
0.8872
Lineage[T.Second]                   -0.177500 0.2681586 342 -0.66192
0.5085
Lineage[T.Third]                     0.221111 0.2681586 342  0.82455
0.4102
Group[T.NEDettol]:Lineage[T.First]   2.275000 0.3792336 342  5.99894
0.0000
Group[T.NEDettol]:Lineage[T.Fourth]  0.955556 0.3792336 342  2.51970
0.0122
Group[T.NEDettol]:Lineage[T.Second]  0.828333 0.3792336 342  2.18423
0.0296
Group[T.NEDettol]:Lineage[T.Third]   0.721667 0.3792336 342  1.90296
0.0579
  Correlation:
                                     (Intr) Gr[T.NED] Lng[T.Frs]
Lng[T.Frt]
Group[T.NEDettol]                   -0.613
Lineage[T.First]                    -0.613  0.500
Lineage[T.Fourth]                   -0.613  0.500     0.500
Lineage[T.Second]                   -0.613  0.500     0.500      0.500
Lineage[T.Third]                    -0.613  0.500     0.500      0.500
Group[T.NEDettol]:Lineage[T.First]   0.434 -0.707    -0.707     -0.354
Group[T.NEDettol]:Lineage[T.Fourth]  0.434 -0.707    -0.354     -0.707
Group[T.NEDettol]:Lineage[T.Second]  0.434 -0.707    -0.354     -0.354
Group[T.NEDettol]:Lineage[T.Third]   0.434 -0.707    -0.354     -0.354
                                     L[T.S] L[T.T]
Grp[T.NEDttl]:Lng[T.Frs]
Group[T.NEDettol]
Lineage[T.First]
Lineage[T.Fourth]
Lineage[T.Second]
Lineage[T.Third]                     0.500
Group[T.NEDettol]:Lineage[T.First]  -0.354 -0.354
Group[T.NEDettol]:Lineage[T.Fourth] -0.354 -0.354  0.500
Group[T.NEDettol]:Lineage[T.Second] -0.707 -0.354  0.500
Group[T.NEDettol]:Lineage[T.Third]  -0.354 -0.707  0.500
                                     Grp[T.NEDttl]:Lng[T.Frt]
G[T.NED]:L[T.S
Group[T.NEDettol]
Lineage[T.First]
Lineage[T.Fourth]
Lineage[T.Second]
Lineage[T.Third]
Group[T.NEDettol]:Lineage[T.First]
Group[T.NEDettol]:Lineage[T.Fourth]
Group[T.NEDettol]:Lineage[T.Second]  0.500
Group[T.NEDettol]:Lineage[T.Third]   0.500                    0.500

Standardized Within-Group Residuals:
         Min          Q1         Med          Q3         Max
-2.47467771 -0.75133489  0.06697157  0.67851126  3.27449064

Number of Observations: 360
Number of Groups:
           Dish Disk %in% Dish
              3              9