Hi there,

I have design a experiment, which will last for 8 weeks. At the
beginning of each week, e.g., Monday, I add some substrate into the experimental unit, and then measured a response. Now, I get a data set,
which is composed of variable of response, and time.

Now, I hope to test the effects of time on response. What's method I
should used? ANOVA is the first method I try. However, I don't think the response is independent, after all, I add the substrate into the
experimental unit every week.

Any helps will be really appreciated. Thanks in advance.

Best,
Jinsong

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On Thu, 12 Dec 2019 13:20:36 +0800, Jinsong Zhao <jszhao@yeah.net>
wrote:

Hi there,

I have design a experiment, which will last for 8 weeks. At the
beginning of each week, e.g., Monday, I add some substrate into the >experimental unit, and then measured a response. Now, I get a data set,
which is composed of variable of response, and time.

Now, I hope to test the effects of time on response. What's method I
should used? ANOVA is the first method I try. However, I don't think the >response is independent, after all, I add the substrate into the
experimental unit every week.

Any helps will be really appreciated. Thanks in advance.

Please note that "ANOVA" in general specifies a variety of
methods dealing with "variance". Another name is "method of
least squares." The other usual method is "maximum likelihood"
(ML). For the simplest designs, the analyses are sometimes
exactly the same in computation (thus, result).

For some designs, the difference in "error measurement" leads to
slight differences in computations. ML methods are sometimes
easier to generalize to complex circumstances, which is mainly
why some computer programs for statisitics use ML.

As to your question about what method ... What outcome
do you expect?

I don't know what you refer to as "substrate" so I only guess
that you might expect a uniform growth from week to week.
Or you might expect an exponential growth.
Or you might expect an early growth, followed by plateau.

In any case: This is a "repeated measure" on some unit.
Is there just one unit? If so, that places a limit - you don't
have the degrees of freedom for error in order to do a
simple test between "weeks", as you might be tempted
to do if there were multiple units. For multiple units, you
want something called "repeated measures."

My suggestion of the hypotheses suggests that the test
would be for linear trend, as opposed to all "non-linear"
components - 1 d.f. for linear, the rest for non-linear.

"Linear trend" would make use of the actual "size" or whatever
at each week, rather than a "response" computed as a
difference or a ratio. If an exponential growh is expected,
the analysis might start with the scores for each week
after the baseline as its ratio over the previous week.

--
Rich Ulrich

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On 2019/12/13 6:42, Rich Ulrich wrote:

On Thu, 12 Dec 2019 13:20:36 +0800, Jinsong Zhao <jszhao@yeah.net>
wrote:

Hi there,

I have design a experiment, which will last for 8 weeks. At the
beginning of each week, e.g., Monday, I add some substrate into the
experimental unit, and then measured a response. Now, I get a data set,
which is composed of variable of response, and time.

Now, I hope to test the effects of time on response. What's method I
should used? ANOVA is the first method I try. However, I don't think the
response is independent, after all, I add the substrate into the
experimental unit every week.

Any helps will be really appreciated. Thanks in advance.

Please note that "ANOVA" in general specifies a variety of
methods dealing with "variance". Another name is "method of
least squares." The other usual method is "maximum likelihood"
(ML). For the simplest designs, the analyses are sometimes
exactly the same in computation (thus, result).

For some designs, the difference in "error measurement" leads to
slight differences in computations. ML methods are sometimes
easier to generalize to complex circumstances, which is mainly
why some computer programs for statisitics use ML.

As to your question about what method ... What outcome
do you expect?

I don't know what you refer to as "substrate" so I only guess
that you might expect a uniform growth from week to week.
Or you might expect an exponential growth.
Or you might expect an early growth, followed by plateau.

In any case: This is a "repeated measure" on some unit.
Is there just one unit? If so, that places a limit - you don't
have the degrees of freedom for error in order to do a
simple test between "weeks", as you might be tempted
to do if there were multiple units. For multiple units, you
want something called "repeated measures."

My suggestion of the hypotheses suggests that the test
would be for linear trend, as opposed to all "non-linear"
components - 1 d.f. for linear, the rest for non-linear.

"Linear trend" would make use of the actual "size" or whatever
at each week, rather than a "response" computed as a
difference or a ratio. If an exponential growh is expected,
the analysis might start with the scores for each week
after the baseline as its ratio over the previous week.

Thank you very much for the explanation and the suggestions. We have 3
units in the experiment. In the experiments, we added a tracers into the
soil, and measured the tracers in fungi. At the beginning of each week,
we added the same amount of tracers into the soil. Finally, we got the following data:

week fungi unit
1 0.70 1
1 0.96 2
1 0.71 3
2 0.92 1
2 0.91 2
2 0.92 3
3 1.50 1
3 2.13 2
3 2.07 3
4 1.88 1
4 1.58 2
4 1.86 3
5 3.83 1
5 4.06 2
5 3.97 3
6 6.82 1
6 5.91 2
6 6.56 3
7 10.05 1
7 10.38 2
7 8.12 3
8 11.17 1
8 11.29 2
8 11.11 3

It's some kind of "repeated measure". We have try to test the effects of
time on the residual of tracers in fungi by linear mixed model in R. The results are:

summary(lme(fungi ~ week, random = ~1|unit, data = xxw))

Linear mixed-effects model fit by REML
Data: xxw
AIC BIC logLik
90.95309 95.31726 -41.47655

Random effects:
Formula: ~1 | unit
(Intercept) Residual
StdDev: 3.439275e-05 1.328719

Fixed effects: fungi ~ week
Value Std.Error DF t-value p-value
(Intercept) -2.489643 0.5977476 20 -4.165041 5e-04
week 1.566310 0.1183717 20 13.232133 0e+00
Correlation:
(Intr)
week -0.891

Standardized Within-Group Residuals:
Min Q1 Med Q3 Max
-1.65241557 -0.80463759 -0.06303065 0.87239442 1.43407057

Number of Observations: 24
Number of Groups: 3

and

anova(lme(fungi ~ week, random = ~1|unit, data = xxw))

numDF denDF F-value p-value
(Intercept) 1 20 282.5119 <.0001
week 1 20 175.0893 <.0001

However, we don't have the full confidence about the results...

Thanks again for your kindly help.

Best,
Jinsong

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On Fri, 13 Dec 2019 16:24:40 +0800, Jinsong Zhao <jszhao@yeah.net>
wrote:

On 2019/12/13 6:42, Rich Ulrich wrote:

On Thu, 12 Dec 2019 13:20:36 +0800, Jinsong Zhao <jszhao@yeah.net>
wrote:

Thank you very much for the explanation and the suggestions. We have 3
units in the experiment. In the experiments, we added a tracers into the >soil, and measured the tracers in fungi. At the beginning of each week,
we added the same amount of tracers into the soil. Finally, we got the >following data:

week fungi unit
1 0.70 1
1 0.96 2
1 0.71 3
2 0.92 1
2 0.91 2
2 0.92 3
3 1.50 1
3 2.13 2
3 2.07 3
4 1.88 1
4 1.58 2
4 1.86 3
5 3.83 1
5 4.06 2
5 3.97 3
6 6.82 1
6 5.91 2
6 6.56 3
7 10.05 1
7 10.38 2
7 8.12 3
8 11.17 1
8 11.29 2
8 11.11 3

It's some kind of "repeated measure". We have try to test the effects of
time on the residual of tracers in fungi by linear mixed model in R. The >results are:

summary(lme(fungi ~ week, random = ~1|unit, data = xxw))

Linear mixed-effects model fit by REML
Data: xxw
AIC BIC logLik
90.95309 95.31726 -41.47655

Random effects:
Formula: ~1 | unit
(Intercept) Residual
StdDev: 3.439275e-05 1.328719

Fixed effects: fungi ~ week
Value Std.Error DF t-value p-value
(Intercept) -2.489643 0.5977476 20 -4.165041 5e-04
week 1.566310 0.1183717 20 13.232133 0e+00
Correlation:
(Intr)
week -0.891

Standardized Within-Group Residuals:
Min Q1 Med Q3 Max
-1.65241557 -0.80463759 -0.06303065 0.87239442 1.43407057

Number of Observations: 24
Number of Groups: 3

and

anova(lme(fungi ~ week, random = ~1|unit, data = xxw))

numDF denDF F-value p-value
(Intercept) 1 20 282.5119 <.0001
week 1 20 175.0893 <.0001

However, we don't have the full confidence about the results...

I don't know the vocabulary of your science, so I'm
still missing the meaning of those numbers. Growth?

I don't see any test done for difference between Units 1-3.
There's probably no notable difference, but I would check.
The Repeated Measures can also be set up as a two-way ANOVA,
with Week by Unit as factors, to get that test.

I suggest that you look at the table rearranged for easier
understanding; and graph the numbers. Then study the
growth curves for each unit.
Here is your data as I table it (pardon any typos, and
variable-font spacing). (Decimal place shifted, for readability.)

70 96 71
92 91 92
150 213 207
188 158 186
383 406 397
682 591 656
1005 1038 812
1117 1129 1111

I would still look at linear trend and "difference from
linear". I see a report of a large correlation, which would
be a measure of Linear; but I don't see a test on it. On the other
hand, your simple growh is of large enough magnitude that
you don't really need to show that test, "Yes, Growth exists
and is large across the 8 weeks."

The test for "nonlinear" will probably be significant if you
use those raw scores and their exponential increase. The test
for Nonlinear would be more interesting on data tested after
taking the logarithms, where it would tend to capture the
effect of the plateaus.

Tukey offered a rule of thumb about transformations -
if your largest numbers are 10 times the smallest, there's
a good chance that you should transform. You have that.

You also have the first four weeks of scores ranging from
70 to 186 -- 116 "points" of range; the next four ranging
from 383 to 1129, or 646 points. A sizable difference.

On the other hand, each 4-week part shows a similar
increase near 3-fold, suggesting that taking logs is linearizing

What I see in the table looks like growth with plateaus -
(1,2); (3,4); 5; 6; (7,8).

Could this be something with spurts in a growth/reproduction
cycle of a bit less than two weeks? - so that there is a split
between 5 and 6?


You have not stated any hypotheses, either Main or
Alternative. If you are going to write a narrative that
will leave you "full confidence in the results", you really
want to have something concrete to say. Start by churning
out whatever may be a set of interesting hypotheses

--
Rich Ulrich

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