GAM is a very useful tool. I hope my suggestions below are useful to help get some good information your data. 

What are the units measured in the dependent variable (population density)? What questions are you asking? Are you interested in interactions or identifying different groupings of your organism?

A bit of data exploration and thinking about the reasons any potential trends may be useful here. Since your model does not check for interactions, plotting the model provides a simple output of a plot for each variable. By default this includes a rug plot of each variable (these are the vertical lines that stick up from the x axis like a rug). This enables you to easily see what data points exist for each variable and something about how they are distributed.

Perhaps the most interesting looking GAM curve fitted is for S (salinity) which has a large dip fitted between around 30 and 100 units of salinity. Is that dip real? Look at the rug plot, there are just three or so data points near 30 and then no data until maybe just one point at over 100. So the confidence interval (shaded area) is very wide. Again is the dip real - well you need to be cautious. The trend is largely related to the three or so data points around 30 appearing to slope steeply downward with a steep slope and then the high salinity data point being back up high. This could just be coincidence or possible explanations to explore are that there is actually a sharp decline as salinity reaches some threshold just over 20 units of salinity - and perhaps the data point at over 100 is a data error or a different species/variant that has a very different threshold for salinity - that might be interesting for you or simply a point that you remove to allow more detailed observation of the points of real interest.

The graphs are all hard to observe trends in because of the wide y axis limits. This is due to the automatic selection of y axis limits due to that big fitted dip in salinity. So try replotting with e.g.

plot(mod_gam2, pages = 1, residuals = T, pch = 19, cex = 0.25,

scheme = 1, col = '#FF8000', shade = T, shade.col = 'gray90', ylim = c(-250, 200))

Make sure the ylim range contains all the data points.

Some of the variables look to be linear or perhaps no discernible relationship (this can be checked by looking at the output (not provided above) from:

summary(mod_gam2)

(in the output, non-statistically-significant variables with have a P-value > say 0.05 and approximately straight line relationships might have an estimated degrees of freedom (edf) of 1).

The model can be re-specified removing the s() from the apparently linear variables, assessing the results again and then considering dropping the variable with the highest P-value. and continuing to drop variables in a stepwise fashion until only "useful" explanatory variables remain. The adjusted R squared from the output can be something of a guide of how good the model is at explaining variation after each change you make.

But fix those issues of checking the single high salinity data point and the wide y axis limits first.

And be cautious of apparent trends that are based on just a few data points.

Hope it helps make some sense of the data.

without being able to answer your question, here are some documents that could be useful :

http://people.bath.ac.uk/sw283/mgcv/

a very complete book:

Mixed Effects Models and Extensions in Ecology with R (I have a PDF version that I can put you at disposal)

alain

GAM is a very useful tool. I hope my suggestions below are useful to help get some good information your data. 

What are the units measured in the dependent variable (population density)? What questions are you asking? Are you interested in interactions or identifying different groupings of your organism?

A bit of data exploration and thinking about the reasons any potential trends may be useful here. Since your model does not check for interactions, plotting the model provides a simple output of a plot for each variable. By default this includes a rug plot of each variable (these are the vertical lines that stick up from the x axis like a rug). This enables you to easily see what data points exist for each variable and something about how they are distributed.

Perhaps the most interesting looking GAM curve fitted is for S (salinity) which has a large dip fitted between around 30 and 100 units of salinity. Is that dip real? Look at the rug plot, there are just three or so data points near 30 and then no data until maybe just one point at over 100. So the confidence interval (shaded area) is very wide. Again is the dip real - well you need to be cautious. The trend is largely related to the three or so data points around 30 appearing to slope steeply downward with a steep slope and then the high salinity data point being back up high. This could just be coincidence or possible explanations to explore are that there is actually a sharp decline as salinity reaches some threshold just over 20 units of salinity - and perhaps the data point at over 100 is a data error or a different species/variant that has a very different threshold for salinity - that might be interesting for you or simply a point that you remove to allow more detailed observation of the points of real interest.

The graphs are all hard to observe trends in because of the wide y axis limits. This is due to the automatic selection of y axis limits due to that big fitted dip in salinity. So try replotting with e.g.

plot(mod_gam2, pages = 1, residuals = T, pch = 19, cex = 0.25,

scheme = 1, col = '#FF8000', shade = T, shade.col = 'gray90', ylim = c(-250, 200))

Make sure the ylim range contains all the data points.

Some of the variables look to be linear or perhaps no discernible relationship (this can be checked by looking at the output (not provided above) from:

summary(mod_gam2)

(in the output, non-statistically-significant variables with have a P-value > say 0.05 and approximately straight line relationships might have an estimated degrees of freedom (edf) of 1).

The model can be re-specified removing the s() from the apparently linear variables, assessing the results again and then considering dropping the variable with the highest P-value. and continuing to drop variables in a stepwise fashion until only "useful" explanatory variables remain. The adjusted R squared from the output can be something of a guide of how good the model is at explaining variation after each change you make.

But fix those issues of checking the single high salinity data point and the wide y axis limits first.

And be cautious of apparent trends that are based on just a few data points.

Hope it helps make some sense of the data.

Dear Alain

Thanks a lot.

IF IT IS POSSIBLE I would like to ask you to send me a copy of that book

my Email: [email protected]

Regards

Mohammad

Dear Tim

Thanks a lot for your useful information.

In fact, the measure unit for environmental factors was the standard one. for example the measure of Salinity was ppt or the measure of Temperature was Centigrade degree. In addition, the measurement for population of Cladophora glomerata macroalgae was wet weight.

I tried to use P value for validation of each variable and figured out that Oxygen and Phosphate were the most important factors contributed in distribution of this species. Therefore I tried to draw 3D response plot with these two factor (see the attachment 1) and it was very useful and easy to interpret. Thanks.

Now here is my main question:

I tired to draw Salinity plot individually (Please see the attachment 2). X axis is the amount of Salinity during study period. now here is the question. What is Y axis? what is it represent? what dose "s(S,1.13) mean in the figure? It is obvious the salinity would not exceed to 60 or 80 or even 100 amounts (As you see all points are in range of 0 to 25 in the plot). so why the plot exceeded to these amounts?

Thanks, yes the effects can become clearer as you simply the model. I would still be cautious around those apparent patterns in areas where there is very little data.

On your main question now with attachment 2: On the plot of S, the Y axis is the "component smooth" so rather than representing wet weight of your organism it is centered around zero and is plotting the partial residuals from the fit - it is useful for checking the residuals and the general shape of the curve. This example is not great as there is a single very high value and lots of variation in response around the median value of S. Looks like one or two residuals may be missing from the yaxis limits?). What you are after is the predicted organism weight based on the x axis.

Brief interlude: the function plot() in R means different things to different objects. If the object is a linear model (lm) or in your case a generalized additive model (gam), then plot prints out something that makes good sense for those objects (in the case of lm it prints out a series of 4 plots to check residuals - in the case of gam it plots the component smooth for each component of the model also allowing residual spread to be checked). 

You want the predicted organism weight according to salinity. So you could do something like this (it seems a bit complex when you want a "simple" output but the plots depend on setting the other variables so best to do that explicitly):

# Create scale for salinity using min and max of S scale from cg data frame

# In this new data frame include sensible (e.g. median) values for the other variables 

salinity.frame 

Dear Tim

Thanks a lot

It hepled

Hi Mohammad,

If you've not got the book (Mixed Effects Models and Extensions in Ecology with R) yet, let me know. I can send you. Additionally, this book might be particularly helpful for GAM:  Wood S.N. (2006) Generalized Additive Models: An Introduction with R. Chapman and Hall/CRC Press.

Although it's hard to tell without having a good idea of experimental design and data, I was wondering why don't you use just GLM? 

Best,

Halim

Dear Halim

I would be very grateful if you  send me a copy of (Mixed Effects Models and Extensions in Ecology with R) (my email: [email protected])

I would like to examine new methods or new investigations to find a best model between environmental data and macroalgae wet weight. Do you new any better statistical method?

Regards

Mohammad

Dear Mohammed and Halim

GAM is very similar to GLM but can cope with non-linear relationships. GLM is a good method if the relationships are linear (we can see from GAM plots that some do not seem to be) or if you can transform non-linear relationships into linear (the GAM results may help identify appropriate transformations). GLM would struggle if the relationships are not able to be transformed well. So actually GAM might be more valid if there are really some complex relationships (as might be expected for some of the variables) - but there are other analyses that may be suitable.

The issue that has not been addressed as yet is potential interaction between the variables. If this was my research I would produce some more exploratory plots e.g. by a lattice plot of the relationships between each variable and perhaps progress to PCA analysis to see if there are any useful patterns.

As Halim says, much will depend on your experimental design. SOme examples: Given the wide range in salinity etc with just a few points at very high salinity, it does not look like this is a controlled environment study (e.g. tanks) but is rather data collected from the "field". Is the fresh weight the weight of one plant and a set number of plants or is it the weight of all plants in a certain area (the method will affect interpretation of results). Do you know anything of taxonomic differences (species, phenotypes, osmotic related compounds etc) between the samples? Are there differences in percent fresh weight between the samples? 

Hope that helps

Tim

Dear Tim

Thanks a lot

I checked the data. I had a mistake in measuring salinity. a point "." was displaced. (107.0 changed to 10.7). As you said I collected data from field work. but weight unit was wet weight which calculated in lab.

Regards

Mohammad

That's great to catch that. That's one of the reasons to do some plots first because mistakes are easy to make. Now the gam on salinity makes good sense.

For fresh weight was there a number of plants per sample or was it all plants for a measured area?

Dear Tim

20×20 cm quadrat (0.04 m2) was performed on rocky substrate to sample from macroalgae communities with three replicate. we picked up every macroalgae in quadrat and preserved it. after that, we measured wet weight in lab.

Dear Mohammed

That's a suitable measurement. So the unit that you might present your results in is kg per square metre. 

Looks like good choice for analysis might be mixed model analysis (this allows the three replicates in each area to be treated as the random component and the different ). Package nlme, function lme.

Still do exploratory plots. At the very least you can get a reasonable idea of the relationship with salinity from your data. You might also see that perhaps phosphorus and silicate could be log transformed to improve the model. And check the response variable (macroalgae measurement ), maybe that can also be transformed (to overcome the effect of a few very large values from a clumped distribution) - this may help in seeing clearer relationships with the environmental variables.

Hope that helps

Tim

Dear Tim

Thanks a lot

It helped very much.

Mohammad,

only the first factor, salinity, has a significant effect on the response. The smooth terms have shown that there is no effect on the response as they were represented by a horizontal line perpendicular to the response axis.

To approcpve that, you may re-fit your model with only the salinity factor, you will approximately obtain the same results with respect to partial t- distribution test.

Excellent answer from Tim Jenkins explaining a lot of the confusion regarding GAM plots.

I suggest also checking this file for a nice introduction of GAM in R: www3.nd.edu/~mclark19/learn/GAMS.pdf

www3.nd.edu/~mclark19/learn/GAMS.pdf

Dear Fernando Zampieri

Thanks a lot

I read the PDF. That was very helpful. But now I have some new question. I found that Salinity and Temperature had significant effect on Macrobenthic communities. Assume that salinity has liner effect and temperature has nonliner effect. How I can find the mathematical equation? I mean something like y=ax+b for salinity and y=ax^2+bx+c for temperature, for example. In these equations "y" is mactobenthic communities and "x" is salinity and temperature. Is this equation is achievable in GAM?

Thanks