def count_empty(index, key):
    """
    Count for any key how many papers leave the value unspecified.
    Note that this function only checks the first of the rows corresponding to a paper.
    """
    count = 0
    for paper, rows in index.items():
        value = rows[0][key]
        if value in {"", "none given", "not given", "blank", "unclear"}:
            count += 1
    return count