Experimental designs that efficiently embed a fixed effects treatment structure within a random effects design structure typically require a mixed-model approach to data analyses. Although mixed model software tailored for the analysis of two-color microarray data is increasingly available, much of this software is generally not capable of correctly analyzing the elaborate incomplete block designs that are being increasingly proposed and used for factorial treatment structures. That is, optimized designs are generally unbalanced as it pertains to various treatment comparisons, with different specifications of experimental variability often required for different treatment factors. This paper uses a publicly available microarray dataset, as based upon an efficient experimental design, to demonstrate a proper mixed model analysis of a typical unbalanced factorial design characterized by incomplete blocks and hierarchical levels of variability.

The choice and optimization of experimental designs for two-color microarrays have been receiving increasing attention
[

Two-color
systems such as spotted cDNA or long oligonucleotide microarrays involve
hybridizations of two different mRNA samples to the same microarray, each of the
two samples being labeled with a different dye (e.g., Cy3 or Cy5; Alexa555 or
Alexa647). These microarrays, also
simply referred to as arrays or slides, generally contain thousands of probes
with generally a few (≤4) spots per probe, and most often just one spot per probe.
Each probe specifically hybridizes to a matching mRNA transcript of interest within
each sample. After hybridization,
microarray images are scanned at two different wavelengths as appropriate for
each dye, thereby providing two different fluorescence intensity measurements for
each probe. Upon further preprocessing
or normalization [

An
increasingly unifying and indisputable message is that the heavily used common
reference design is statistically inefficient [

The
intent of this review is to reemphasize the use of mixed models as the
foundation for statistical analysis of efficient factorial designs for
microarrays. Mixed model analysis for
microarray data was first proposed by Wolfinger et al. [

Efficient experimental designs are
typically constructed such that their factors can be broadly partitioned into
two categories:

These design structure or random
effects factors are typically further partitioned into two subcategories:

Currently, there is much software available for microarray data analysis, some of which is only suited for studies having only a treatment structure but no pure design structure. Common examples include the analysis of data generated from single channel systems (e.g., Affymetrix) or of log ratios generated from common reference designs. When no random effects are specified, other than the residuals, the corresponding statistical models are then simply fixed-effects models. Ordinary least squares (OLS) inference is then typically used to infer upon the treatment effects in these studies. OLS is appropriate if the assumption is valid that there is only one composite residual source of variability such that the residuals unique to each observation are NIID.

Conversely,
statistical analysis of efficient two-color experiments having a fully
integrated treatment and design structure needs to account for fixed and random
effects as typical of a

Recall
that some designs may be characterized by different levels of variability
thereby requiring particular care in order to properly separate biological from
technical replication, for example. Hence, it is imperative for the data analyst
to know how to correctly construct the hypothesis test statistics, including
the determination or, in some cases, the estimation of the appropriate degrees
of freedom. Although, some of these issues have been
discussed for balanced designs by Rosa et al. [

Optimized interwoven loop design for 9 treatments using R package SMIDA (Wit et al., 2005). Each circle represents a different treatment group. Each arrow represents a single array hybridization with circle base representing the Cy3 labeled sample and tail end representing the Cy5 labeled sample.

Even
for some balanced factorial designs, where the standard errors for comparing
mean differences for levels of a certain factor are the same for all pairwise
comparisons, the experimental error structure can vary substantially for
different factors. That is, substantial
care is required in deriving the correct test statistics, particularly with
split plot arrangements [

Zou et al. [_{2} solution (M). Unfoliated leaves from three to
four plants were drawn and pooled for each treatment at each of three different
times after postinoculation; 2, 8, and 24 hours. Hence, the treatment structure was comprised
of a

A graphical depiction of the 13
hybridizations that superimposes the design structure upon one replicate of the

Experimental design for one replicate from
Zou et al. (2005). Treatments included a
full _{2} (M) control inoculate and numbers
indicating time (2, 8, or 24 hours) after inoculation. Each arrow represents a
single array hybridization with circle base representing the Cy3 labeled sample
and tail end representing the Cy5 labeled sample. Solid arrows refer to the A-loop
design of Landgrebe
et al. (2006).

We
arbitrarily consider gene expression measurements for just one particular gene
based on the GEO submission from Zou et al. [

For the purposes of this review, we
concentrate our attention just on the subdesign characterized by the solid arrows
in Figure

It should be somewhat apparent from the
A-loop design of Figure

The
complex nature of different levels of replication in the A-loop this design is further
confirmed in the classical analysis of variance or ANOVA [

Classical ANOVA of log intensities for duplicated
A-loop design component of Figure

Source | Mean square | Expected mean square | ||
---|---|---|---|---|

Inoculate | ||||

Time | ||||

Inoculate^{*}time | ||||

Dye | ||||

Rep(inoculate^{*}time) | ||||

Array(time) | ||||

Error |

^{*}Sums of squares.

^{†}Degrees of freedom.

^{‡}^{*}time) have the same expected mean square and

Classical
ANOVA is based on equating sums of squares (SS), also called quadratic forms,
to their expectations; typically this involves equating mean squares (MS),
being SS divided by their degrees of freedom (ν), to their EMS. For completely balanced designs, there is
generally one universal manner in which these quadratic forms, and hence the
ANOVA table, are constructed
[

Table

Consider,
for example, the test for the main effects of inoculate denoted as Factor A in
Table ^{*}time) in Table ^{*}time) is said
to be the denominator or ^{*}time) defines the
experimental unit or the biological replicate for inoculate effects. Hence, the correct ^{*}time in Table ^{*}time
and rep(inoculate^{*}time) share the same EMS such that the correct

It was previously noted from the
A-loop design of Figure ^{*}time)
and arrays(time) such that marginal mean comparisons between the three times,
2, 8, and 24 hours, will be affected by more noise than marginal mean comparisons
between the three inoculates which were directly and indirectly connected
within arrays.

Note
that under the null hypothesis of no time effects

In
our example, consider the synthesized
^{*}time), array(time), and residual. With reference to (

To
help further illustrate these concepts, let us conduct the ANOVA on the data
generated from the A-loop design of Figure ^{*}time
interaction, the appropriate F-test statistic is

Classical
ANOVA of log intensities for duplicated A-loop design component of
Figure

Type 3 analysis of variance | ||||||||
---|---|---|---|---|---|---|---|---|

Source | DF^{†} | Sum of squares | Mean square | Expected mean square | Error term | Error DF | Pr > F^{†} | |

Trt | 2 | 0.7123 | 0.3561 | Var(Residual)
+
1.5
Var(sample(inoc^{*}time)) + Q(inoc,inoc^{*}time) | MS(sample(inoc^{*}time)) | 6 | 3.13 | 0.1172 |

Time | 2 | 3.7737 | 1.8868 | Var(Residual)
+
2
Var(sample(inoc^{*}time)) +
2Var(array(time))
+ Q(time,inoc^{*}time) | 1.3333
MS(array(time)) + 1.3333 MS(sample(inoc^{*}time))
| 13.969 | 3.27 | 0.0683 |

Inoc^{*}time | 4 | 0.6294 | 0.1573 | Var(Residual)
+
1.5
Var(sample(inoc^{*}time)) + Q(inoc^{*}time) | MS(sample(inoc^{*}time)) | 6 | 1.38 | 0.3435 |

Dye | 1 | 0.0744 | 0.0744 | Var(Residual) + Q(dye) | MS(Residual) | 5 | 2.19 | 0.1989 |

Rep(inoc^{*}time) | 6 | 0.6826 | 0.1137 | Var(Residual)
+
1.5
Var(sample(inoc^{*}time)) | MS(Residual) | 5 | 3.35 | 0.1030 |

Array(time) | 12 | 4.3330 | 0.3610 | Var(Residual) + 1.5 Var(array(time)) | MS(Residual) | 5 | 10.63 | 0.0085 |

Residual | 5 | 0.1699 | 0.0339 | Var(Residual) | . | . | . | . |

^{†}Degrees of freedom.

^{‡}

SAS code for classical ANOVA and EGLS
inference. Comments describing purpose
immediately provided after corresponding code between /^{*} and ^{*}/ as with a regular
SAS program. EGLS based on REML would
simply involve substituting

title “Mixed model analysis of log fluorescence intensity data from gene 30”;

data=gene30 /^{*} name of data
as provided in Table ^{*}/

method = type3;

/^{*} Provides
classical ANOVA table and EGLS based on ANOVA estimates of VC ^{*}/

/^{*} If REML
estimates of VC are desired, change above line to method = reml; ^{*}/

where ((array <=

/^{*} Using A-loop
component (arrays 1-9, 15-23) of Table ^{*}/

class rep array inoc time dye;

/^{*} name of fixed
and random classification factors in design ^{*}/

model ly = inoc time
inoc^{*}time dye

/^{*} Specify
response variable and fixed effects here ^{*}/

/ddfm = kr

/^{*} Use
Kenward-Roger's procedure to estimate denominator degrees of freedom ^{*}/

e3;

/^{*} e3 will print
the contrast matrices KA, KB and KAB (see (

(^{*}/

random array(time) rep(inoc^{*}time)
; /^{*}
Specify random effects ^{*}/

estimate “k1 contrast”

int ^{*}time

/^{*} contrast
coefficients as specified for k1 in (^{*}/

estimate “k2 contrast”

int ^{*}time

/^{*} contrast
coefficients as specified for k2 in (^{*}/

The
synthesized denominator

Although the classical ANOVA table
is indeed instructive in terms of illustrating the different levels of
variability and experimental error, it is not the optimal statistical analysis method
for a mixed effects model, especially when the design is unbalanced. A mixed-model
or GLS analysis more
efficiently uses information on the design structure (i.e., random effects) for
inferring upon the fixed treatment structure effects [

Unfortunately,
GLS, in spite of
its optimality properties, is generally not attainable with real data because
the VC (e.g.,

Recall
that with unbalanced designs, quadratic forms are not unique such that ANOVA
estimators of VC will not be unique either. Nevertheless, type III quadratic
forms are most commonly chosen as then the SS for each term is adjusted for all
other terms, as previously noted. Although
ANOVA estimates of VC are unbiased, they are not efficient nor optimal in terms
of estimates having minimum standard error [

Once
the VCs are
estimated, they are substituted for the true unknown VCs to provide the
“estimated” GLS or EGLS of the fixed
effects. It is important to note that typically EGLS = GLS for balanced designs,
such that knowledge of VC is somewhat irrelevant for point estimation of treatment
effects. However, the same is generally not true for unbalanced designs, such
as either the A-loop design derived from Figure

As
we already intuitively noted from the A-loop design of
Figure

Now, the denominator degrees of
freedom for inference on these two contrasts should also differ given that the
nature of experimental error variability somewhat differs for inoculate
comparisons as opposed to time comparisons as noted previously from
Figure

Contrasts
are also used in EGLS to provide ANOVA-like

Now the interaction between
inoculate and time is a

The
EGLS statistics used for testing the overall importance of these main effects
or interactions are approximately distributed as

Unfortunately,
much available software used for mixed model analysis of microarray data does
not carefully take into consideration that various fixed effects terms of
interest may have different denominator degrees of freedom when constructing

It
was previously noted that the estimated standard errors for EGLS on two
contrasts

The overall EGLS tests for ID_REF #30 for testing
the main effects of inoculate, time and their interaction as based on the previously
characterized complementary contrasts are provided separately for ANOVA versus
REML estimates of VC in Table

EGLS inference on overall importance of fixed
effects for ID_REF #30 based on REML versus ANOVA (type III quadratic forms)
for estimation of variance components using output from SAS PROC MIXED (code in
Figure

Type 3 tests of fixed effects using REML | Type 3 tests of fixed effects using ANOVA | ||||||
---|---|---|---|---|---|---|---|

Effect |
Num DF^{*} |
Den DF^{*} | F value |
Pr > F^{†} |
Den DF^{*} | F value |
Pr > F^{†} |

Inoc | 2 | 5.28 | 3.12 | 0.1273 | 6.36 | 3.48 | 0.0954 |

Time | 2 | 17.8 | 2.81 | 0.0870 | 22.8 | 3.27 | 0.0563 |

Inoc^{*}time
| 4 | 5.28 | 1.26 | 0.3893 | 6.36 | 1.38 | 0.3392 |

Dye | 1 | 5.43 | 2.27 | 0.1879 | 5.15 | 2.19 | 0.1973 |

^{*}Num Df = numerator degrees of freedom; Den DF = denominator
degrees of freedom.

^{†}

Dataset for ID_REF #30 for all
hybridizations (14 arrays/loop x2 loops) in
Figure

Obs | array | inoculate | time | rep | dye | y | ly |
---|---|---|---|---|---|---|---|

1 | 1 | R | 2 | 1R2 | Cy3 | 16322.67 | 13.9946 |

2 | 1 | M | 2 | 1M2 | Cy5 | 20612.48 | 14.3312 |

3 | 2 | M | 2 | 1M2 | Cy3 | 10552.21 | 13.3653 |

4 | 2 | S | 2 | 1S2 | Cy5 | 10640.89 | 13.3773 |

5 | 3 | S | 2 | 1S2 | Cy3 | 24852.98 | 14.6011 |

6 | 3 | R | 2 | 1R2 | Cy5 | 21975.92 | 14.4236 |

7 | 4 | R | 8 | 1R8 | Cy3 | 30961.96 | 14.9182 |

8 | 4 | M | 8 | 1M8 | Cy5 | 13405.08 | 13.7105 |

9 | 5 | M | 8 | 1M8 | Cy3 | 13103.51 | 13.6777 |

10 | 5 | S | 8 | 1S8 | Cy5 | 15659.44 | 13.9347 |

11 | 6 | S | 8 | 1S8 | Cy3 | 20424.47 | 14.3180 |

12 | 6 | R | 8 | 1R8 | Cy5 | 34244.92 | 15.0636 |

13 | 7 | R | 24 | 1R24 | Cy3 | 15824.29 | 13.9499 |

14 | 7 | M | 24 | 1M24 | Cy5 | 13014.05 | 13.6678 |

15 | 8 | M | 24 | 1M24 | Cy3 | 17503.11 | 14.0953 |

16 | 8 | S | 24 | 1S24 | Cy5 | 27418.99 | 14.7429 |

17 | 9 | S | 24 | 1S24 | Cy3 | 37689.16 | 15.2019 |

18 | 9 | R | 24 | 1R24 | Cy5 | 55821.64 | 15.7685 |

19 | 10 | S | 2 | 1S2 | Cy3 | 28963.28 | 14.8219 |

20 | 10 | S | 8 | 1S8 | Cy5 | 38659.44 | 15.2385 |

21 | 11 | S | 8 | 1S8 | Cy3 | 41608.78 | 15.3446 |

22 | 11 | S | 24 | 1S24 | Cy5 | 41844.79 | 15.3528 |

23 | 12 | R | 2 | 1R2 | Cy3 | 12132.41 | 13.5666 |

24 | 12 | R | 8 | 1R8 | Cy5 | 19131.53 | 14.2237 |

25 | 13 | R | 8 | 1R8 | Cy3 | 31067.04 | 14.9231 |

26 | 13 | R | 24 | 1R24 | Cy5 | 26197.03 | 14.6771 |

27 | 14 | N | 2 | 1N2 | Cy3 | 18540.91 | 14.1784 |

28 | 14 | M | 2 | 1M2 | Cy5 | 24971.88 | 14.6080 |

29 | 15 | R | 2 | 2R2 | Cy3 | 9612.25 | 13.2307 |

30 | 15 | M | 2 | 2M2 | Cy5 | 9212.11 | 13.1693 |

31 | 16 | M | 2 | 2M2 | Cy3 | 10322.23 | 13.3335 |

32 | 16 | S | 2 | 2S2 | Cy5 | 10979.19 | 13.4225 |

33 | 17 | S | 2 | 2S2 | Cy3 | 8061.40 | 12.9768 |

34 | 17 | R | 2 | 2R2 | Cy5 | 6737.37 | 12.7180 |

35 | 18 | R | 8 | 2R8 | Cy3 | 8807.09 | 13.1044 |

36 | 18 | M | 8 | 2M8 | Cy5 | 8696.95 | 13.0863 |

37 | 19 | M | 8 | 2M8 | Cy3 | 15186.20 | 13.8905 |

38 | 19 | S | 8 | 2S8 | Cy5 | 23477.49 | 14.5190 |

39 | 20 | S | 8 | 2S8 | Cy3 | 19424.30 | 14.2456 |

40 | 20 | R | 8 | 2R8 | Cy5 | 18198.99 | 14.1516 |

41 | 21 | R | 24 | 2R24 | Cy3 | 19630.00 | 14.2608 |

42 | 21 | M | 24 | 2M24 | Cy5 | 15629.14 | 13.9320 |

43 | 22 | M | 24 | 2M24 | Cy3 | 10875.49 | 13.4088 |

44 | 22 | S | 24 | 2S24 | Cy5 | 20816.21 | 14.3454 |

45 | 23 | S | 24 | 2S24 | Cy3 | 24647.70 | 14.5892 |

46 | 23 | R | 24 | 2R24 | Cy5 | 22148.96 | 14.4350 |

47 | 24 | S | 2 | 2S2 | Cy3 | 17795.09 | 14.1192 |

48 | 24 | S | 8 | 2S8 | Cy5 | 34569.11 | 15.0772 |

49 | 25 | S | 8 | 2S8 | Cy3 | 44175.28 | 15.4310 |

50 | 25 | S | 24 | 2S24 | Cy5 | 38020.46 | 15.2145 |

51 | 26 | R | 2 | 2R2 | Cy3 | 34689.07 | 15.0822 |

52 | 26 | R | 8 | 2R8 | Cy5 | 62219.10 | 15.9251 |

53 | 27 | R | 8 | 2R8 | Cy3 | 22724.21 | 14.4719 |

54 | 27 | R | 24 | 2R24 | Cy5 | 19594.71 | 14.2582 |

55 | 28 | N | 2 | 2N2 | Cy3 | 11755.32 | 13.5210 |

56 | 28 | M | 2 | 2M2 | Cy5 | 12599.55 | 13.6211 |

In
completely balanced designs, ANOVA and REML lead to identical estimates of VC and
identical EGLS inference, provided that all ANOVA estimates of VC are positive. ANOVA estimates of VC that are negative are
generally constrained by REML to be zero, thereby causing a “ripple” effect on
REML estimates of other VC and subsequently on EGLS inference [

Recent
work on the optimization and comparison of various efficient microarray designs
have been based on the assumption of OLS inference; that is, no random sources
of variability other than residuals are considered
[

Secondly,
almost all of the design optimization work has been based on the use of Cy3/Cy5
log ratios as the response variables rather than dye-specific log intensities as
used in this review. This data
reduction, that is, from two fluorescence intensities to one ratio per spot on
an array, certainly eliminates array as a factor to specify in a linear model. However, the use of log ratios can severely
limit estimability and inference efficiency of certain comparisons. Suppose that instead of using the 36 log intensities
from the duplicated A-loop design from arrays 1–9 and 15–23 of
Table _{2}Cy3 and Cy5 fluorescence intensities for array 1 from
Table

The
relative efficiency of some designs may be seen to depend upon the relative
magnitude of biological to technical variation [

There
are a number of different criteria that might be used to choose between
different designs for two-color microarrays. We have already noted that the
interwoven loop design in Figure

For
one particular type of optimality criterion, Landgrebe et al.
[

Recall
that Figure

One should perhaps compare two
alternative experimental designs having the same factorial treatment structure,
but a different design structure, for contrasts of highest priority, choosing
those designs where such contrasts have the smaller standard error. Let us consider the following comparisons:

Now
the comparison of efficient designs for the relative precision of various
contrasts will generally depend upon the relative magnitude of the random
effects VC as noted recently by Hedayat et al. [

Shrinkage
or empirical Bayes (EB) estimation is known to improve statistical power for
inference on differential gene expression between treatments in microarray
experiments [

We
have provided an overview of the use of mixed linear model analysis for the
processing of unbalanced microarray designs, given that most efficient incomplete
block designs for microarrays are unbalanced with respect to various
comparisons. We strongly believe that much mixed-model software currently
available for the analysis of microarrays does not adequately address the
proper determination of error terms and/or denominator degrees of freedom for
various tests. This would be
particularly true if we had chosen to analyze all of the data for ID_REF #30 in
Table

We
believe that it is useful to choose proven mixed-model software (e.g., SAS) to
properly conduct these tests and, if necessary, to work with an experienced
statistician in order to do so. We have concentrated our attention on the
analysis of a particular gene. It is, nevertheless, straightforward to use SAS
to serially conduct mixed-model analysis for all genes on a microarray
[

Any mixed model,
including that specified in (

Again,
the distributional assumptions on the random and residual effects are specified
the same as in the paper but now written in matrix algebra notation:

Once
the VC are estimated, they are substituted for the true unknown VC in

It
was previously noted in the paper that the mean difference

Now,
when the VC are known, these two contrasts can be estimated by their GLS,

Kenward
and Roger [

_{ij.}

Recall that the main effects of
times (Factor B) involves a joint test of

Recall that the interaction between
the effects of inoculates and times was

The
EGLS test statistic for testing the overall importance of any fixed effects
term, say

Support from the Michigan Agricultural Experiment Station (Project MICL 1822) is gratefully acknowledged.