## Reciprobit distribution

How to plot reaction times on a reciprobit scale.

# Reaction time distribution

Subjects are notoriously bad at reacting as fast as possible in a consistent fashion. This is readily apparent from the considerable trial-by-trial variation in reaction times despite standardized experimental conditions (see e.g. the Saccade Introduction tutorial, which shows saccade latencies faster than 250 ms but also slower than 500 ms). Furthermore, the distribution of reaction times is typically skewed, with a tail towards long reaction times. A histogram of manual reaction times, that were gathered for a single subject responding to a sound modulation change (data can be found here), easily shows this.


%% Histogram
rt = RT.easy; % let's take the reaction times (ms) for the "easy" task
x  = 0:50:1000; % reaction time bins (ms)
N  = hist(rt,x);  % count reactions in bin
N  = 100*N./sum(N); % percentage

figure(1) % graphics
h  = bar(x,N); % histogram/bar plot
% Making figure nicer to look at
set(h,'FaceColor',[.7 .7 .7]); % setting the color of the bars to gray
box off; % remove the "box" around the graph
axis square;
xlabel('Reaction time (ms)'); % and setting labels is essential for anyone to understand graphs!
ylabel('Probability (%)');
xlim([0 1000]);

% optional, save figure
print('-depsc','-painter',[mfilename '1']);


# Transformation - Promptness

Often, simply taking the inverse of the reaction times (promptness) leads to a nicely symmetric, well-behaved Gaussian distribution. You can see that the inverse reaction time histogram (grey bars) corresponds nicely to a Gaussian distribution (black lines) with the same mean and standard deviation as the actual distribution in the figure below.



%% Inverse reaction time
rtinv  = 1./rt; % inverse reaction time / promptness (ms-1)

n    = numel(x); % number of bins in reaction time plot
x    = linspace(1/2000,0.01,n); % promptness bins
N    = hist(rtinv,x);
N    = 100*N./sum(N);

figure(2)
h = bar(x*1000,N);
hold on
set(h,'FaceColor',[.7 .7 .7]);
box off
axis square;
xlabel('Promptness (s^{-1})');
ylabel('Probability (%)');
title('Reciprocal time axis');
set(gca,'YTick',0:5:100,'XTick',0:1:8);
xlim([0 8]);

% Does this look like a Gaussian?
% Let's plot a Gaussian curve with mean and standard deviation from the
% promptness data
mu  = mean(rtinv);
sd  = std(rtinv);
y  = normpdf(x,mu,sd);
y  = y./sum(y)*100;
plot(x*1000,y,'ks-','LineWidth',2,'MarkerFaceColor','w');

% optional, save figure
print('-depsc','-painter',[mfilename '2']);

And you don't even have to believe your own eyes: you can test this with a one-sample kolmogorov-smirnov test.


% And test it with a one-sample kolmogorov-smirnov test
% Since this test compares with the normal distribution, we have to
% normalize the data, which we can do with zscore, which is the same as:
% z = (x-mean(x))/std(x)
[h,p]  = kstest(zscore(rt)); % for reaction time
if h
str    = ['The null hypothesis that the reaction time distribution is Gaussian distributed is rejected, with P = ' num2str(p)];
else
str    = ['The null hypothesis that the reaction time distribution is Gaussian distributed is NOT rejected, with P = ' num2str(p)];
end
disp(str); % display the results in command window
[h,p]  = kstest(zscore(rtinv)); % for inverse reaction time
if h
str    = ['The null hypothesis that the inverse reaction time distribution is Gaussian distributed is rejected, with P = ' num2str(p)];
else
str    = ['The null hypothesis that the inverse reaction time distribution is Gaussian distributed is NOT rejected, with P = ' num2str(p)];
end
disp(str); % display the results in command window


# Cumulative probability

Now, histograms are problematic in the sense that their shape depends on the bin size and that they have an effectively arbitrary vertical scale. Cumulative histograms are normalized (running from 0 to 1 / 100%), can represent all data without binning (continuous), and facilitate comparisons between data sets (see further below). In the figure below, quantiles at 1, 2, 5, 10, 20, 30, ..., 70, 80, 90, 95, 98, and 99% are plotted (red-faced circles) together with the actual data.


%% Cumulative probability
% Normalized scale and nicer shape
x = sort(1000*rtinv);
n = numel(rtinv); % number of data points
y = 100*(1:n)./n; % cumulative probability for every data point
figure(3)
plot(x,y,'k.');
hold on

% Now, plot it cumulative probability in quantiles
% this is easier to compare between different distributions
p    = [1 2 5 10:10:90 95 98 99]/100; % some arbitrary probabilities
q    = quantile(rtinv,p); % instead of hist, let's use quantiles

h = plot(q*1000,p*100,'ko','LineWidth',2,'MarkerFaceColor','r');
hold on
xlabel('Promptness (s^{-1})');
ylabel('Cumulative probability (%)');
title('Cumulative probability plot');
box off
axis square;
set(gca,'YTick',0:10:100,'XTick',0:1:8);
xlim([0 8]);
legend(h,'Quantiles','Location','SE');

% optional, save figure
print('-depsc','-painter',[mfilename '3']);


# Probit scale

The cumulative Gaussian distribution results in a straight line by using a probit scale for the ordinate. You can convert cumulative probability data in Matlab as follows:


cdf     = q;
myerf       = 2*cdf - 1;
myerfinv    = sqrt(2)*erfinv(myerf);
chi         = myerfinv;


which is also done by pa_probit.m. This immediately allows you to visually check (don't believe statistics, see for yourself) whether the distribution is actually Gaussian (otherwise, the data will not yield a straight line), the 50% point indicates the median of the distribution, and the standard deviation is directly related to te slope of the line. To adhere to the convention that long reaction times are on the right, the abscissa scale is also mirrored (data multiplied by -1).


%% Probit
figure(4)
% raw data
x = -1./sort((rt)); % multiply by -1 to mirror abscissa
n = numel(rtinv); % number of data points
y = pa_probit((1:n)./n); % cumulative probability for every data point converted to probit scale
plot(x,y,'k.');
hold on

% quantiles
p    = [1 2 5 10:10:90 95 98 99]/100;
probit  = pa_probit(p);
q    = quantile(rt,p);
q    = -1./q;
xtick  = sort(-1./(150+[0 pa_oct2bw(50,-1:5)])); % some arbitrary xticks

plot(q,probit,'ko','Color','k','MarkerFaceColor','r','LineWidth',2);
hold on
set(gca,'XTick',xtick,'XTickLabel',-1./xtick);
xlim([min(xtick) max(xtick)]);
set(gca,'YTick',probit,'YTickLabel',p*100);
ylim([pa_probit(0.1/100) pa_probit(99.9/100)]);
axis square;
box off
xlabel('Reaction time (ms)');
ylabel('Cumulative probability');
title('Probit ordinate');

% this should be a straight line
x = q;
y = probit;
b = regstats(y,x);
h = pa_regline(b.beta,'k-');
set(h,'Color','r','LineWidth',2);

% optional, save figure
print('-depsc','-painter',[mfilename '4']);