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It’s possible to fit a linear regression when one of the variables takes discrete values, however, the simple scatterplot produced by this kind of dataset is often not optimal:

One option is to add some random noise (“jitter”) to the discrete values to make the distribution of those values more clear. Note that jitter is applied only to the scatterplot data and does not influence the regression line fit itself:

A second option is to collapse over the observations in each discrete bin to plot an estimate of central tendency along with a confidence interval:

The simple linear regression model used above is very simple to fit, however, it is not appropriate for some kinds of datasets. The Anscombe’s quartet dataset shows a few examples where simple linear regression provides an identical estimate of a relationship where simple visual inspection clearly shows differences. For example, in the first case, the linear regression is a good model:

The linear relationship in the second dataset is the same, but the plot clearly shows that this is not a good model:

In the presence of these kind of higher-order relationships, lmplot() and regplot() can fit a polynomial regression model to explore simple kinds of nonlinear trends in the dataset:

A different problem is posed by “outlier” observations that deviate for some reason other than the main relationship under study:

In the presence of outliers, it can be useful to fit a robust regression, which uses a different loss function to downweight relatively large residuals:

When the y variable is binary, simple linear regression also “works” but provides implausible predictions:

The solution in this case is to fit a logistic regression, such that the regression line shows the estimated probability of y = 1 for a given value of x :

Note that the logistic regression estimate is considerably more computationally intensive (this is true of robust regression as well) than simple regression, and as the confidence interval around the regression line is computed using a bootstrap procedure, you may wish to turn this off for faster iteration (using ci=None ).

An altogether different approach is to fit a nonparametric regression using a lowess smoother . This approach has the fewest assumptions, although it is computationally intensive and so currently confidence intervals are not computed at all:

“Animals have very different sensory, perceptive abilities to us. In some cases they’re better and in some cases they’re worse, but in all cases they’re different,” says Matthew Savoca at the NOAA Southwest Fisheries Science Center in Monterey, California.

One explanation is that animals simply mistake plastic for familiar food items – plastic pellets, for example, are thought to resemble tasty fish eggs. But as humans we are biased by our own senses. To appreciate animals’ love of plastic, scientists must try to view the world as they do.

Humans are visual creatures, but when foraging many marine animals, including albatrosses , rely primarily on their sense of smell. Savoca and his colleagues have conducted experiments suggesting that some species of nike air max 90 prestashop
and are attracted to plastic by its odour. Specifically, they implicated dimethyl sulfide (DMS), a compound air jordan 11 original release
, as the chemical cue emanating from plastic. Essentially, algae grows on floating plastic, and when that algae is eaten by krill – a major marine food source – it releases DMS, attracting birds and fish that then munch on the plastic instead of the krill they came for.

Even for vision, we can’t jump to conclusions when considering the appeal of plastic. Like humans, marine turtles rely primarily on their vision to search for food. However, they are also thought to possess the capacity to see UV light, making their vision quite different from our own.

Qamar Schuyler at The University of Queensland, Australia, has got into turtles’ heads by modelling their visual capabilities and then measuring the visual characteristics of plastics as turtles see them. She has also examined the stomach contents of deceased turtles to get a sense of their preferred plastics. Her conclusion is that while young turtles are relatively indiscriminate, older turtles preferentially target soft, translucent plastic. Schuyler thinks her results confirm a long-held idea that turtles mistake plastic bags for delicious jellyfish.

Colour is also thought to factor into plastic consumption, although preference varies between species. Young turtles prefer white plastic, while Schuyler and her colleagues found that seabirds called shearwaters opt for red plastic.

Besides sight and smell, there are other senses animals use to find food. Many marine animals hunt by echolocation, notably toothed whales and dolphins. Echolocation is known to be incredibly sensitive, and yet dozens of sperm whales and other toothed whales have been found dead with stomachs full of plastic bags, car parts and other human detritus. Savoca says it’s likely their echolocation misidentifies these objects as food.

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