What statistics test can I use to test for the following example:
Let's say I have time trend data on one group of subjects.  The number
of individuals with a certain characteristic increases from 32 to 40
to 41 to 46 (these are counts and not means, so there is no standard
deviation).  I want to know if that represents an increase over time
that is different from no increase.  In other words, is the slope of
that curve significantly different from zero.  How would I go about
testing this?  How do I compare slopes?

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Clarification of Question by qetuo-ga on 29 Jan 2006 18:19 PST

The characteristic is binary I guess (yes or no)
There are four time points: 
Time point 1: 32 yeses
Time point 2: 40 yeses
Time point 3: 41 yeses
Time point 4: 46 yeses

Sample size 78

Testing to see whether there is a trend and whether these values are
different from each other;  one problem is that the yeses are
cumulative; so the trend can only stay at 32 or increase from 32 --> I
believe this makes chi-square invalid?

Hi, some questions:
-What measurement level has the characterisit (metrical, ordinal, nominal, binary)
-What do you mean by 32 to 40 and 41 to 46? Is the number not certain
at each timepoint?
-Do you have two or more timepoints?

Hi,

I would probably use the uncertainty on a slope which you may easily
find on the internet.  Then find the range of slope using an
approriate confidence level

But, before calculating, you should organize the data in a more simple
way. Something like the proportion of new 'yeses' for each Time point.
 Also, look for a way to weight each point because one with a larger
nbr of answers is probably more accurate.

Hope this helps

If you only have 4 single data points, you cannot do much with
statistical inferencing. Ideally, you should have a large set of data
points in each sample, in order to compare across samples. In fact,
it's impossible to test whether these numbers are different from each
other based on what you have.

What you can do is regress the number of yesses on time, that is,
essentially plotting the points in 2-space with time on x-axis and
yeses on y-axis. The line that is closest vertically to all the points
will give you a very rough idea of the relationship between number of
yesses and time.

Numerically, the equation you are trying to estimate is Y=a+bX, where
b will give you the slope of this line, which is the effect on number
of yesses as time increases. Your estimate of b will be

[(X1-Xbar)(Y1-Ybar)+(X2-Xbar)(Y2-Ybar)+(X3-Xbar)(Y3-Ybar)+(X4-Xbar)(Y4-Ybar)]/
[(X1-Xbar)^2+(X2-Xbar)^2+(X3-Xbar)^2+(X4-Xbar)^2]

Intuitively this is the variation of X and Y (covariance of x and y)
over the variation in X only (variance of x), which gives you the
variation in Y per unit of change in X.

Your intercept a will be Ybar-b*Xbar.

These are the most efficient linear unbiased estimators of your simple
linear regression.

To test whether b is statistically significantly different from 0, or
whether the relationship between time and number of yesses is real
(which it obviously is, by looking at the data), you will need the
variance of b. This is equal to

sigma^2/[(X1-Xbar)^2+(X2-Xbar)^2+(X3-Xbar)^2+(X4-Xbar)^2], where
sigma^2 is your sample variance and Xbar the mean of X. Take the
square root of this quantity to get the standard error of b.

b divided by the above quantity should give you a test statistic for
the t distribution with 2 degrees of freedom. You should reject or
accept accordingly the hypothesis that time has a significant effect
on number of yesses.

And there you have it. Of course, this is much easier with a software.
If you have a TI, there should be a function under the stat menu that
lets you input x and y values after which it will spit out all the
relevant statistics you need.