A simple statement in line graphs stating that constant slope indicates declining relative performance??

Unlikely to work as the first impression is created by the visual chart. The only solution use log scale only

Would there be any situation where a linear scale would work as the major feature of a line graph is to show shape over time.

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And you use of the logarithmic scale here is…interesting. I see what you mean here, but I’m not sure of the point. A constant slope on a log graph, for example, would require exponential growth of the whatever, which is almost certainly unsustainable.

Examples with real data would be useful here.

Here are a few examples:

1) Body Temperature. Tylenol undoubtedly would have incredibly robust data in patients with fevers but the magnitude of the effect would be small. A graph that showed temperature based on F (or even C) using an absolute 0 would not be very impressive. But the meaningful range would probably only be from 95 deg to 105 deg.

2) Blood glucose. Normal mice have blood glucose values of ~160 mg/dL. The variability if very low with a coefficient of variation of ~5 %. Given that the intrinsic variability of the measure is low, then an increase of 20 % would be highly statistically significant (and functionally quite ill), but it would not be very “impressive” if you require someone to show a “0” on the graph.

3) Body weight. A body weight of 0 is irrelevant and therefore should not be included on a graph. If a group of mice (or people) start out weighing 40g (or 200 lb) and they all lost 5–10 % that would be very significant and very functionally important. But graphing it with a 0 would not suggest signficance. I have seen a lot of data where investigators show BW graphs in ways that UNDEREMPHASIZE parameters that are highly regulated. They see a 5 % weight loss and think is means nothing, but in fact it can be quite important.

Two other examples to consider … Again Temperature … We use Farenheit and Celcius, each of which have their own meaning of 0. You might extend your argument to say the only real measure of temperature is Kelvin and therefore all data should add 273 to it. In truth, Celcius and Farenheit just defined new 0 values based on practical experience. That is what one does if they cut an axes short.

Another example to consider would be time … what is 0 time? Going back to the big bang? Birth of Christ? 2009? Again, one usually selects at “time 0” that is relevant for the data being presented.

In both cases, as I think should be argued for scientific data, the person presenting the data needs to be fair about what they are presenting. Data with variances should be different than that without. There should be some consideration of RELEVANCE of the magnitude of the effect. However, I do agree that often popular media overlook this aspect of data presentation.