I guess the problem you are running into, is that you do not have a reliable (enough) measure of the parameter-values in the two parental populations (hence, the large within-group variation). Of course you could easily compare the cross-bred observed parameter-values to fixed values of the parental populations, for instance using two simple t-tests comparing to the population-value rather than 0. However, this would mean that you simply discard the uncertainty that you know there is, thereby running the serious risk of overstating results.

In other words: if you cannot show that the parameter values from the cross-breds differs significantly from each of the parental distributions, than either there is no such difference or you do not have enough (reliable) data to support that claim.

But, perhaps I'm missing a clever technique ...

You can see this with Box plots of each group, shown side-to-side.

Hi Ivan,

I would try bootsrapped confidence intervals of the mean for each of your 3 groups.

You could easily check if 95% CI of F1'mean lie within parents'means confidence intervals.

You can do it for instances with R boot package, or with R ez package (more friendly)

Good luck

Pierre

I will use the null hypothesis where you want to evaluate that your mean is not significantly different than the average of your two other means. However, this will not always occur, but it will be quickly to evaluate. The other alternative is to use some order statistics. Here, you can evaluate that o(1) is significantly different thatn o(2) and so on. But you need to assume a preliminary distribution (normal?) and then do all the math to construct this hypothesis.

I think it would be interesting to see 3 histograms of the data, side by side or stacked on top of each other with the different pops coded with different colors or fills. That would show how they are different. Plus the histogram would show the spread of the data which seems important.

This distributíon is normal?

You can use One of The groups as control?

Dear Ivan

Did you get your measurements in a sort of experimental design such as CRD or RCB with independent replicates? If so, run an appropriated ANOVA model and use a mean separation technique such as LSD, Duncan, or Tukey - having in mind that the conservativeness follows LSD < Duncan < Tukey. Alternatively, you could build confidence interval to show some descriptive statistics of location for the parents and the progenies.

Hope this helps.

Hi,

I do like the bootstrap idea, but perhaps you can do something more directly related to your hypothesis, i.e., do the cross-bred values lie between the two parents. The test statistic is the percentage of times the cross-bred value is between the two parents. Then construct a sampling distribution by bootstrapping the offspring values and noting the number of times they lie between the parents.

Simplest way to create a simple graph with the mean values on the Y-axis, the 3 groups on the X-axis. Each group had a vertical line = 95% C.I. This would show your group falls between since it would be the middle one on the X-axis. In addition, C.I. that do not overlap would reveal significant differences. Visual presentations are always easier to interpret than numeric results in a table.

This is a common problem in certain areas, and the acronym TOST (two one-sided tests) is often used. I would under no circumstances do two t-tests. They would not be consistent. Properly formulated, this is a one way ANOVA with 3 levels. You wish to test 2 specific contrasts, each in a 1-sided test. This is not difficult. It can be done easily in SAS or R with designed contrasts.

If I understood you correctly, you want to compare the values of F1 against an expected values rather than against the values of the parental populations...

If you want to test the hypothesis that the values of a quantitative variable for a single group is (statistically) different than an "expected" value, just do an one-sample t-test (if you data meet the assumptions of the test, of course).

How it would be in R:

#Hypothetical values to be tested

values

Isn't the idea her to do a regression of the mid-parental value versus the F1 value? It sounds like your F1 value will be equal, or nearly so, to the mid-parent value. If so, the regression line would have a slop of 1, and that would show that the F1 was between the parents....

I don't know why my "e"s were cut off. (here, and slope, not her and slop!). ha!

Analysis of group means with unequal sample sizes lets differences in standard deviations be measured against degrees of freedom values other than n-1

Ivan, the answers from Paul Thompson and Mohsen Mohammadi+Eduardo Lourenço, 'specially designed' ANOVA (using contrasts) and post-hoc multiple comparisons test after ANOVA 'adapted' to use the F1 as control, seemed to better deal w/ your question (if you data is normally distributed).

Have you already plotted the boxplots of parental populations and F1? it would tell you much of their distributions and rough significance of the medians. You can see if notches superpose or not.

Your assumption is that population B lies between A and C, that is, A

I think u have also compare the post hoc multiple comparison, it Iuse to compare the single factor, with One way ANOVA model use a mean separation technique like LSD (Low scored deviation ) and HSD (High Scored deviation), Tukey or SNK test - denotes different significant levels or letters superscript. Those Stasticall was anaysis by SPSS, Mintab, etc.

Best of Luck

Hope this helps.

The test you can use depends on the nature of parameter measured (response variable). If a continuous variable, a chi square test would be appropriate.

Hi,

In your design, subjects are breeding couples. Note that if you have several offspring per breeding couple, this is a kind of repeated measure.

Pay attention, if you work at population level, you loose completely parents-offspring(s) relationships. This mix up sources of variances.

What Stephan Arndt suggested sounds perfect. Compare each offspring to his own parent, not to the population. You can obtain p-value through permutation (don't resample), I think it is better than bootstrap. In the case where a breeding couple has several offspring, use each breeding couple as random factor.

For ploting you data, you could try something more meaningfull than population means. Here is a raw idea. For each breeding couple and each offspring you rescale traits value of everyone in a way that first parent has 0, second parent has 1. (Note that offspring value may or not lie in that rescaled range, this is exactly what Stephan Arndt suggested). Then you could get bootstrapped confidence interval of the offspring's mean value (rescaled) and ckeck if CI includes or not 0 or 1.

About regression analysis, it could be of interest to have a look to effect of parents values (or delta in parents value) on offspring value. I don't know your study species but you could have for instance (1) two parents having both large values who have offspring with large value and (2) one parent with low and one parent with high value who have offspring with high value. --> Further than offspring value lie in parents value, such deeper investigation might indicate that large value for the studied trait is dominant over smaller value. I am not genetician at all, but I would have a look in this direction.

Cheers,

Pierre

P-value is not good idea forever!

Just try to model using GLM (GLMM) with a suitable link, and a set of dummy variables (contrasts)

Model 1: A, B, F1 (enough 2 dummy variables)

Model 2: {AB} F1

Model 3: A {B F1}

Model 4: B {A F1}

Model 5: {A B F1} - analogue H0

And ranked them based on a posterior model weights (BIC (CAIC ...)

Use evidence ratio wi/wj

Good luck!

Hi Ivan,

Please tell me more about this parameter that you considering as response variable. In your statistical analysis. Could you send me the sample design that your are using to evaluate the null hypothesis? I addition i would like to ask you about if you considered that the respose variable fitted in the normal distribution?

best regards,

Mario

If you use a Bayesian ANOVA approach, you could compute the posterior probability that the ordering is the predicted ordering. This is not very difficult to do: You might try lmer/arm or the BayesFactor packages in R.

Hi all, thank you for the many productive responses. Sorry for not getting back immediately with more detail. The variable measured was a measure of body mass that we call fueling capacity (for migratory birds, the maximum body mass reached relative to the baseline - minimum - body mass. So in fact it is a percentage but it can take values of more than 1 in case the bird gets more then 100% its minimum body mass. In any case it is continuous but never can get less than 0). What we did was to run a GLMM model using group (parental groups) as a fixed factor and nest as a random factor. Then we performed Tukey post hoc tests. They showed that the two parental populations are significantly different from each other but the F1 was not significantly different from the parents. Graphically, however, they are pretty much in between the parents. The within-group variability is probably too big to detect differences between groups. However, I don't know if this approach is really what we want. I have an idea that I am not sure is "legal": calculate an "expected value" as the average value between the two parents, and then see in what measure the offspring deviate from this value. E.g.:

Parents have values of 1.0 and 0.5

Expected value: 0.75

F1 values: 0.6; 0.8; 0.65

Deviations from exp. values: -0.15; 0.05; -0.1

Then I would test the hypothesis that these values are not significantly different than 0. Any feedback on this? Thanks a lot folks, it has already been very useful, but I'm waiting for more help!! Cheers,

Ivan

Another look at the data & model.

What if you just use a regression or ANCOVA (separate slope): log weight offspring - response, the log-weight of mother & father - predictors, plus a group of ... as factor/

What about central limit theorem?