Just to clarify: what do you define exactly as "proportion"? Because, strictly speaking, it makes think about categorical variables on which ANOVA and T-tests are in principle not to be performed.

Assuming you measure a continuous variable, then two different kind of questions arise:

- are the ANOVA assumptions met (normality, homoscedasticity, independance...) ?

- what do you exactly want to proove ? Figuring out your graph, my first idea would be that you want to show an increase with age, and in that case linear regression would probably work very well. But if you want to proove only a change with age, ANOVA can be suited Other specific comparisons will answer to other questions.

So, as often, you must better precise your question ("significance between age groups" does not mean much...) to have suited answers...

I understand that within each age group you have multiple measurements of binary outcomes (inhibitory and excitatory). Are these measurements independent? Non-independent measurements could be for example measurements at the same location if there are multiple locations within each individual; measurements within the same individual if there are multiple individuals in the same age group.

An other kind of non-independence could be if the same individual is part of multiple age groups (new measurements when the subject gets older).

Depending on the above aspects of the study design a hierarchical and/or repeated measures model may give better estimates than a one-way approach (ANOVA or else), especially in terms of the accuracy of standard errors, widths of confidence intervals and p values.

Another aspect is whether ANOVA is a good approximation for the binary outcome or not. If the sample size within each group is similar and not too small relative to the percentage of inhibitory currents etc. then summarising the binary outcomes into percentages may be a reasonable approximation.

However instead of thinking over how well the ANOVA approximates the question you could use a modelling technique that is designed for binary outcomes, like (a) logistic regression, or (b) Poisson or multinomial regression summing the numbers of currents within each experimental unit.

(How to compare the age groups is a different question.)

either the T-test or ANOVA does not apply to test proportions. familywise multi-test for proportions (such as Z-test or Fisher Exact) should fine here.

A pretty simple thing to do is assume there is no difference between ages. Sum all counts, divide by 6 to get the no difference expected number, then do a chi-square test (a one-way simple contingency table). Becomes more complicated if you want to test for difference between adjacent ages. Another thing that could be done is to fit a simple linear regression with age as the predictor variable and the proportion as the response variable. The significance of the slope coefficient is a trend test. I don't know your data or your actual goal so I felt simple is better than complex.

Hi, everyone! OK, a good try is GLMM model with binomial distribution. You can do this in R, S-Plus, SAS, etc. without problems.

Dear Cherry,

Thanks for your question. You have confused us with the term "proportions". Indeed, it refers to categorical data that would need to be analyzed by the Fischer's Exact test (for dichotomous variables) or by the Pearson's chi-square test (for more than two categories) by crosstabulation, which is obviously not your case.

Based on your attached figures, I understand that you want to compare continuous variables. As suggested by the other colleagues above, please perform a one-way ANOVA. For pairwise comparisons, please use the Bonferroni or the Dunnets correction.

Hope this was helpful.

Best wishes,

Dimitri Raptis

Dear Hui, Bruce, Dimitri,

Cherry has not given the details of the study design, particularly what structure of repeated measures he has. So the Fisher exact test, trend test, one-way ANOVA (or any test that treats the measurements as independent) may give misleading results.

Gabor

Hi,

Others have mentioned the correct answer, e.g., a chi-square or GLMM. A simple alternative is using a logistic regression predicting the outcome (the dependent variable) from age (the independent variable). You can perform an overall test (similar to the overall chi-square), a test for linear trend (by treating age as a continuous variable), or any of a variety of pairwise tests (depending on how you set up your dummy variables). A good resource for this kind of analysis is: Hosmer, D. W., & Lemeshow, S. (1989). Applied Logistic Regression. New York: John Wiley & Sons. (Although there is a more recent edition.)

Steve

Again, to stress what others have mentioned, you cannot receive the correct answer until we know whether your measurements are on the same set of animals as they develop or whether you made the measurements on different sets of animals in each age group. Do the different bars in your graph contain the same set of animals or does each bar contain a different set of animals?

I'll take a stab at this. It seems from your description that each animal has excitatory and inhibitory mini currents (making a total) and you'd like to see if there's a shift in proportion of one of these. I agree with others that this should push you towards a binomial distribution in a generalized linear model of some variety, with % ihibitory=successes, and total (inh+exc) = trials. If you measure this at different ages within the same animal (over time), then this could analyzed as a generalized linear mixed model. In either case, you may want to model age as continuous, rather than as categorical...unless there is a theoretical reason as to why there would be a true step function (e.g. menopause decreasing estrogen as a step for considering cardiovascular risk in women). If model in % as a function of continuous is reasonable, then you could implement a binomial growth model (fixed and random intercepts and slopes). Note, this would not look like a "straight" line in terms of %...rather it would be ogival as it approached 0 and 100%...and the variability would appropriately be asymetrical as you deviated from 50%...with the scale appropriately bounded at 0 and 100%.

These are the key features that should steer you away from ANOVA/t-test, which is perfectly happy to predict negatives and values above 100%.

Another possibility is that inhibitory and excitatory counts could be modeled as a mixture model. As is, in a binomial, you're implicitely assuming there is a total current count determined by some process, of which some will be excitatory and some inhibitory. So, % inhibitory in a binomial model couldn't distinguish between an increase in percent inhibitory because inhibitory currents increased or because excitatory decreased. A mixture of, say 2 Poisson models would allow you to model each. For that matter, in the poor-man's version, you could simply analyze each in independent Poisson models...and merely constrain your assertions appropriately.

Hi, you could use a one way contingency chi square or linear regression analysis.

If there are homogenity in variance, normality in your data and the same number of replicates by treatment, I think that Anova can be a good choice... otherwise, Anova-permutation

http://adn.biol.umontreal.ca/~numericalecology/Rcode/

All in R

To properly answer this question you should first decide on:

- whether your leading variable is continuous (A) or categorical (B), and

- whether your age groups should be considered independent (A) or dependent variables (B)

Depending on the answers to these two questions, you may have several alternative methods:

1A-1B - consider 1w-ANOVA or block design ANOVA; then multiple comparisons tests, f.i. planned Scheffe test of contrasts

1A-2B - consider ANOVA with repeated measures; then multiple comparisons tests, i.e. Connover

1B-2A - consider 2 x c chi2 or conditional logistic regression

1B-2B - consider Cochrane Q test

Thanks for all valuable inputs to my questions!

So guess I should explain my data more. By proportion, I meant the percentage of spontaneous miniature inhibitory postsynaptic currents (mIPSCs) over the total spontaneous miniature currents (inhibitory plus excitatory, mIPSCs + mEPSCs). This is from one single cell. Then for each age, I have 6 to 10 cells and I averaged the percentages for each age and presented in the graph as shown. Each percentage is from a different cell (although the same cell type), so I'm not sure whether this can be considered continuous or categorical? I'm also not sure whether age groups should be considered independent or dependent variables but I think it should be independent (because I expect the percentages of inhibitory currents over total currents depend on age)? Correct me if I'm wrong.

Quick notes

Thanks for the clarification of design. Multiple cells per age selected independently of the cells for other age categories mean that this does not require a repeat measures analysis. How 'age' is measured and whether the age divisions are summarizations versus unavoidable results of the available measurement technique remains unclear. In either case, the logistic regression and GLM approaches each can be viable (depending on how the response percentages are treated:chi square simply compares proportions among age groups, logistic regression is appropriate for modeling the association for predictive or other purposes if these are treated as a binary measures for each cell. I am unclear from your graph whether you enter individual cell responses versus the average across the age category for GLM as you did in your graph.

Chi square test will be the best option to know the significance of difference in proportions in different age groups. If expected count in any cell is less than 10 or if sample size is low, Fischer's exact test or Chi square with Yate's correction should be applied.

Expected count for a cell can be calculated by the formula (Column total / Grant total ) * Row total

In my experience you may use chi square test to compare proportion of inhibitory current among the groups. Otherwise you may compare the mean values of inhibitory corrent of each groups using anova test or kruskal wallis test if your data are not normally distributed. By by

I believe, based on the description, the data is one percentage per cell (binomial), rather than yes/no (binary). Chi-square would be useful if the data were binary: "Did the percentage of CELLS that were in state X increase with age?" This data, however asks: "Did the percent of observed miniature currents that were inhibitory increase within cells increase as a function of age? ...or "Was the percentage of currents that were inhibitory higher in older cells?"

Since it sounds like a single cell would not be measured at multiple ages, then a repeated measures technique for analysis is not necessary. Interestingly, it could be noted that individual cells ARE measured multiple times if you consider each current a measurement...but since these are aggregated into counts the analysis need not allow for nesting at least with regards to age.

The binomial distribution is the one which best approximates your dendent variable, given the information available. Each cell will have multiple miniature currents. Some of those will be inhibitory. Some will be excitatory. If there are no other options, then the binomial is a very good match...with the exception that current type may not be independent from one current to another...for example, it could be that the next current is more likely to be the same as the previous current.

The probelem with using ANOVA and t-test is that this data is, by definition, not normal. It MAY be close enough under a lot of circumstances, but looking at your figures, P3 and P6 are a good ways away from 50%...where binomial data is closest to "normal". Your data will have a positive tail the closer it is to zero and a negative tail the closer it is to 100%. That means your central tendency is being over estimated when below 50% and under estimated when you're above 50%. Variances will also likely be too large and implicitely assumed to symmetrical.

All these factors lead away from statistics dependent on the normal distribution, but not necessarily toward non-parametrics. As I said in my previous post, a generalized linear model for binomial data would very nicely match this data.

Earlier, I'd wrote that you need not worry about nesting age (witin cell). Depending on your specific methology, there may be other sources of correlation that should be modelled, though. For example, were multiple cells measured within the same dish?

If you had cells of different ages on the same dish, and multiple dishes, then it may be that you should allow for within-dish correlation (many options).

Another possibility is that each dish contained cells of only 1 age...and that you had several dishes per age. Here, you may want to consider a hierarchical model treating "dish" as the unit of analysis, within which single cells provided you examples (like children in a classroom).

A "concern" would be if each dish contained only cells of 1 age...and you used only one dish per age. This would represent a potential confound in the experiment that could not be "fixed" statistically.

To what I have understood, the parametrization should be:

Y= Percentage or Proportion mIPSCs/mIPSCs+mEPSCs

X1=Age Group (continuous or categorical upon your needs)

X2=Cell (=individual or subject; random effect)

X1 should be a continuous variable if you want explain results in general terms of age (or model the phenomenon based on age). Conversely, it should be a categorical variable if you want to explain results in terms of age group (this should be desirable, because it is regardless of the relationship – linear or not – between mIPSCs and Age)

To my opinion, logistic regression is the most adequate procedure in this framework. On the contrast, ANOVA or other parametric tests would be incorrect.

Here is one way you might code if were doing in SAS. There are many options to be considered over and above this, but here is a starting point.

assuming we have a dataset with the following variables:

cell - a unique identifier for a particular cell being tested

inhibitory - the count of inhibitory miniature currents

excitatory - the count of excitatory miniature currents

total - the sum of inhibitory and excitatory

Age - a numeric variable in some real units (be that months or passages,etc.)

1) treating age as continuous...no nesting mecessary given methods

*model with age on a continuum, but making point estimates based on model if want to use bar graph...however more representative graph generated later in code;

proc glimmix plots=(residualpanel pearsonpanel) empirical;

class cell;

model inhibitory/total = Age/dist=binomial;

estimate "Age=3" Intercept 1 Age 3,

"Age=6" intercept 1 Age 6,

"Age=9" intercept 1 Age 9,

"Age=12" intercept 1 Age 12,

"Age=15" intercept 1 Age 15,

"Age=18" intercept 1 Age 18

/cl ilink adjdfe=row;

OUTOPUT OUT=Perc_Inh pred(ilink noblup)=Mu_Perc_Inh LCL(ilink noblup)=LCL UCL(ilink noblup)=UCL;

run;

*Make figure overlaying model predictions and data;

data temp_for_figure;

set Perc_Inh;

Percent_Inh = inhibitory/(inhibitory+excitatory ); /*for graphing purposes*/

run;

proc sgplot data=temp_for_figure;

band x=age lower=LCL upper=UCL;

series x=age y=Mu_Perc_Inh;

scatter x=age y =Percent_Inh;

run;

quit;

2) age as categorical

proc glimmix plots=(residualpanel pearsonpanel) empirical;

class cell Age;

model inhibitory/total = Age/dist=binomial;

lsmeans Age/cl adjdfe=row stepdown=logical plots=(meanplot(ilink));

run;

There are methodological considerations that may change the options used in these models...in particular, if there is a hierarchy implied in the methods, the degrees of freedom generated by this model may be more liberal than fully appropriate.

type #1: "OUTOPUT" should be "OUTPUT"

Doh!

also "empirical" option tells SAS to employ classical sandwhich estimation...hopefully this will appropriately adjust for any misspecification in the model...the possibility of autocorrelation between inhibitory or excitatory miniature current observation...

The best test is Z test of proportion between groups in respect of large sample and if it is a small sample the 't' test may be applied . In this we can estimate the population parameters also in confidence interval of proportions. The samples are more than two groups, chi square is the nice test.