How long does it take to read a sentence with X number of characters? How does the time needed to read a sentence scale with the number of characters? Or does this time scaling depend on something more than just character count?

For example, let $X$ be the number of characters and $F(X)$ the seconds to read an entire sentence of $X$ characters long. There are 3 options:


Linear $F(X) \in \Theta(X)$:  Is the relationship linear? That is, if it takes 0.5 seconds to read "hello", it would take 1.0 second to read "hello world": $F(X) = 0.1 X$
Sublinear $F(X) \in o(X)$: Or does it decay? That is, as people read a long sentence, their reading speed increases (crazy example $F(X) = 1 / X$)
Super-linear $F(X) \in \omega(X)$: Or is it like quadratic? That is, each additional character ads more time than the previous word? (example: $F(X) = X ^ 2$)


Which of the above 3 options best describes reading speed based on length of text? Is the specific functional form of this relationship known?



Motivation

I am creating an iPhone app where I need to show transient confirmation messages. For example, when a user submits a comment, I pop up a message saying "thanks for submitting your comment". Shortly after, the message will fade away. There exist many such transient messages all over my iPhone app. 

What I would like to do is to calculate the optimal time to show each message based off of the number of characters in that message. I want users to have enough time to comfortably read the message, but not so long that the message annoys them. The purely UX version of this question is here:

How long does it take to read X number of characters?
I've studied this a little bit within the context of timing responses to personality test items. General models of reading speed look at both the time to read the words as well as to comprehend. From memory, eye tracking studies have shown how the eyes will often back track to confusing parts of a sentence (apologies for lack of reference).

Some general findings that I've found in my own data on time to answer personality test items of different lengths:


Expect an initiation period. I.e., the time between displaying a message and the commencement of reading. In mathematical terms, this will translate to a non-zero constant in your model of reading time as a function of properties of the text.
The more familiar the content and the words are to the reader, the quicker they will be. This translates both in absolute terms that simple words are quicker to read, and also to interactions between individuals and the content.
In general, character count, word count, and syllable count tend to be highly correlated in normal prose. From my own data, these indices tend to have a relatively linear effect on reading speed.
More complex sentences (e.g., ones with negation, multiple clauses, etc.) take longer to read. For example, I found that sentences with "not" took systematically longer to process.
Individuals differ substantially in reading speed. Expect to see a distribution of individual differences that is a little bit positive skewed from normal. Thus, if you are designing a system where you determine the length of time text is displayed, you may need to pitch the speed of your system to capture a certain percentage of that distribution (e.g., 95, 99% etc.). Various factors explain variation in reading speed including whether English is a second language and education.
There are also outliers. People get distracted. Thus, there is reading time when people are focussed and there is reading times that incorporate distraction and other processes.


More generally, if you have the resources, you can do some pilot testing on a sample of individuals where you time how long people take to read the text, ideally within the context of the application. This could allow you to develop a domain specific model of reading speed.

Although a little tangential, some articles that discuss speed of item responding in a personality testing context include Casey and Tyron (2001) and Wagner-Menghim (2002).

References


Casey, M.M. &amp; Tryon, W.W. (2001). Validating a double-press method for computer administration of personality inventory items.. Psychological assessment, 13, 521.
Wagner-Menghin, M. (2002). Towards the identification of non-scalable personality
questionnaire respondents: Taking response time into account.
Psychologische Beiträge, 44(1), 62-77.